11institutetext: Michele Cantiello (✉) 22institutetext: INAF-Astronomical
Observatory of Abruzzo, Via Maggini snc, 64020, Teramo, Italy; 22email:
<EMAIL_ADDRESS>33institutetext: John P. Blakeslee (✉)
44institutetext: NSF’s NOIRLab, Tucson, AZ 85719, USA; 44email:
<EMAIL_ADDRESS>
# Surface Brightness Fluctuations

Michele Cantiello & John P. Blakeslee

###### Abstract

The Surface Brightness Fluctuation (SBF) method is a powerful tool for
determining distances to early-type galaxies. The method measures the
intrinsic variance in a galaxy’s surface brightness distribution to determine
its distance with an accuracy of about 5%. Here, we discuss the mathematical
formalism behind the SBF technique, its calibration, and the practicalities of
how measurements are performed. We review the various sources of uncertainties
that affect the method and discuss how they can be minimized or controlled
through careful observations and data analysis. The SBF technique has already
been successfully applied to a large number of galaxies and used for deriving
accurate constraints on the Hubble-Lemaître constant $H_{0}$. An approved JWST
program will greatly reduce the systematic uncertainties by establishing a
firm zero-point calibration using tip of the red giant branch (TRGB)
distances. We summarize the existing results and discuss the excellent
potential of the SBF method for improving the current constraints on $H_{0}$.

##  0.1 Introduction

Stellar counts have long been used to study unresolved or partially-resolved
stellar populations baum1955 . However, it was not until the development of
CCD astronomy that the statistical characteristics of the discrete nature of
stellar counts allowed for the formulation of the surface brightness
fluctuations (SBF) method ts88 ; tal90 , one of the most reliable and stable
extragalactic distance indicators used within the range of a few to about 150
Mpc.

As the distance to a stellar population increases, individual stars blend
together and the brightness profile becomes smoother, but there remain
statistical fluctuations on small scales because of the discrete nature of the
stars. The SBF method leverages the random nature of stellar counts and
luminosities to measure a quantity that corresponds to the luminosity-weighted
mean brightness of the stars in the stellar population. For evolved stellar
systems, this is approximately equal to the mean brightness of the stars on
the red giant branch (RGB).

The SBF method was initially developed to determine distances of relatively
nearby elliptical galaxies observed from the ground. Over the past few
decades, the technique has undergone improvements in the analysis tools and
benefited from increasingly advanced astronomical facilities. Consequently,
SBF distances are now derived for a wider class of objects beyond ellipticals:
spiral bulges, dwarf galaxies, irregulars, and others. Moreover, the distance
limit has increased by a factor of $\sim$7 from the original predictions, and
will increase further in the coming years.

Quantitatively, what is measured in the SBF analysis is the ratio between the
intrinsic brightness variance (in the absence of blurring) and the average
surface brightness over a specific region of a galaxy. This ratio has the
units of flux and it is equivalent to the ratio between the second and first
moment of the stellar luminosity function within the region analyzed. Although
the definition is simple, there are a number of intricacies in practice. These
involve how to measure the SBF amplitudes in astronomical images, the optimal
targets for the measurement, and how to calibrate SBF magnitudes, either
empirically or using stellar population synthesis.

In the following sections, we provide a detailed description of the method and
summarize the recent results, especially regarding the “Hubble tension.” We
conclude with a discussion of the future prospects for the method.

##  0.2 Characterization of the SBF Signal

The main conceptual difficulty related to SBF does not pertain to the
measurement of fluctuations or their calibration, which we discuss below.
Rather, a cause of confusion, for some, concerns the nature of the indicator
itself. Unlike other commonly used distance indicators like parallaxes,
Cepheid variables, and Type Ia supernovae, the SBF signal is not easily
“visualized”, and it is not immediately apparent how fluctuations in star
counts can be converted into a reliable estimate of distance.

To grasp how the SBF signal is produced and used for distance estimation, we
can consider a simplified model of a CCD image of a stellar population
unblurred by a point spread function (PSF). Let all the stars have the same
(unknown) luminosity $L_{*}$, with $n_{*}$ being the average number of stars
per pixel and $d$ the distance. Then, $f_{*}=L_{*}/(4\pi d^{2})$ is the flux
contributed per star. We cannot directly measure $f_{*}$, but we can measure
the surface brightness $F_{*}$ as the mean flux per pixel, $F_{*}=n_{*}\times
f_{*}$. Since $n_{*}\propto d^{2}$ and $f_{*}\propto 1/d^{2}$, $F_{*}$ is
independent of the distance of the stellar system.

We can also measure the standard deviation in the flux per pixel. Because of
Poisson fluctuations in the numbers of stars, the standard deviation is
$\sigma_{F}=\sqrt{n_{*}}f_{*}$. Now, we define the SBF flux as the ratio of
the variance $\sigma_{F}^{2}$ to the mean:

$\bar{f}=\sigma^{2}_{F}/F_{*}=n_{*}f_{*}^{2}/(n_{*}\,f_{*})=f_{*}=L_{*}/(4\pi
d^{2}).$ (1)

Thus, in this trivial case where the stars are all identical, the SBF flux
reduces to the mean flux per star, which depends inversely on the square of
the distance.

In reality, stars follow a luminosity function, and a more realistic treatment
moresco22 shows that the ratio of the variance to the mean surface brightness
reduces to:

$\bar{f}=\frac{\sum_{i}n_{i}f_{i}^{2}}{\sum_{i}n_{i}f_{i}}\equiv\frac{\bar{L}}{4\pi
d^{2}}\,,$ (2)

where $n_{i}$ is the number of stars of apparent brightness $f_{i}$, and
$\bar{L}$ is the ratio of the second and first moments of the stellar
luminosity function, corresponding to the luminosity-weighted mean stellar
luminosity of the population. For a more detailed derivation of $\bar{f}$, see
cervino08 or Sec. 3.9.1 of moresco22 . $\bar{L}$ is easily calculated from
stellar population models; because of the luminosity weighting, it is
sensitive to the brightest stars in the population. Thus, the SBF flux
$\bar{f}$, or the equivalent SBF magnitude $\overline{m}$, is solely
determined by the stellar luminosity function and the galaxy distance.

##  0.3 From SBF Magnitudes to Distances

Similar to other standard candle methods, deriving SBF distances requires
measuring an apparent magnitude, $\overline{m}$, and then adopting an absolute
$\overline{M}$ based on a calibration relation to obtain the distance modulus,
$\overline{m}{-}\overline{M}$. The following sections describe the process for
measuring $\overline{m}$ and estimating $\overline{M}$, as well as the
characteristics of suitable target galaxies and the preferred passbands for
reliable SBF measurements.

###  0.3.1 SBF measurements step by step

Accurate SBF measurements require correcting and accounting for the different
sources of noise or contamination that afflict the signal. In a sense, the
fluctuation signal itself is a sort of noise in the star counts per pixel; in
the absence of detector noise, photon shot noise, contamination from
background sources, and blurring by the PSF, the SBF would simply be the
Poisson variance among pixels due to the varying number and luminosity of
stars in each pixel, normalized to the local mean flux of the galaxy. However,
all of these effects are present and must be dealt with.

Since the SBF signal is convolved with the PSF, it can be distinguished from
photon and detector noise, which have white-noise power spectra, at least
prior to distortion correction. For this reason, the analysis is done in the
Fourier domain, where the SBF signal is measured by determining the amplitude
of the variance component in the image power spectrum on the scale of the PSF.
Numerous papers have described the steps involved in SBF measurements tal90 ;
jensen98 ; blake99 ; cantiello05 ; mei05iv ; blake10 ; jensen15 ; carlsten19 ;
moresco22 . In general, the procedure can be summarized as follows.

After combining individual dithered exposures to make a cleaned, stacked image
free from cosmic rays, satellite trails, and detector defects, we estimate and
subtract the sky background, mask any bright stars and neighboring galaxies,
then derive a smooth isophotal model of the galaxy surface brightness. This
model is then subtracted from the image, the remaining point-like and extended
sources are masked, and a low-order fit to the background is derived and
subtracted to eliminate large-scale residuals from the galaxy model
subtraction. This masking, modeling and subtraction procedure is typically
iterative. We then detect and mask all compact objects (foreground stars,
globular clusters in the galaxy, faint background galaxies) down to a signal-
to-noise threshold of $\sim\,$4.5 using an automatic photometry program. These
objects, along with any other sources of non-SBF variance (visible dust,
brighter satellite galaxies, tidal features, regions of poor model residuals,
etc.) are masked after the smooth model and large-scale residuals are removed.

The residual masked frame is then normalized by the square root of the model
and contains both the variance from the stellar fluctuations of interest as
well as fluctuations from unexcised sources fainter than the detection limit,
along with the unconvolved photon and detector noise. The next step is to
analyze the power spectrum of the normalized residual masked frame. The
fluctuations from the stellar counts and unexcised sources are convolved with
the instrumental PSF in the spatial domain; hence, they are multiplied by the
Fourier transform of the PSF in the Fourier domain. For accurate SBF
measurements, it is crucial to have an accurate characterization of the local
PSF. This can be achieved using individual point sources in the field or by
utilizing PSF modeling and reconstruction methods.

After azimuthally averaging the power spectrum111The power spectrum is by
definition the squared modulus of the Fourier transform; this explains the
normalization of the residual frame by the square root of the galaxy model.,
the amplitude $P_{0}$ of the astrophysical component of the normalized and
masked residual frame’s power spectrum $P(k)$ is obtained as the constant term
that multiples the “expectation power spectrum” $E(k)$, which is calculated
as: $E(k)=\widetilde{\hbox{PSF}}(k)\otimes\widetilde{\hbox{M}}(k)$, where
$\widetilde{\hbox{M}}(k)$ is the power spectrum of the normalized PSF, and
$\widetilde{\hbox{PSF}}(k)$ is the power spectrum of the mask window function.
There is also the white noise (approximately independent of wavenumber $k$)
component $P_{1}$; thus the power spectrum is modeled as: $P(k)=P_{0}\times
E(k)+P_{1}$.

If the large-scale light distribution is well-subtracted, the power spectrum
will be well represented by these two components over most the $k$ range.
Additional power may exist at low $k$ (largest spatial scales) due to
imperfect galaxy subtraction. The power may also be suppressed at some values
of $k$ due to correlation of the pixel noise resulting from geometric
distortion correction; this is especially problematic when using linear
interpolation of the pixel values. However, for sinc-like interpolation
kernels cantiello05 ; mei05iv , the “damage” to the power spectrum is confined
to high $k$ (the smallest scales). In this case, the highest and lowest
wavenumbers can simply be omitted from the fit range to ensure an accurate
measurement.

It is crucial to note that the fitted $P_{0}$ value includes contamination
from unexcised sources fainter than the detection limit. Any astrophysical
source of fluctuation that is not generated from the stellar counts must be
estimated and removed by subtracting the contribution from $P_{0}$. The
presence of dust in some galaxies may also affect the power spectrum, and
hence the fitted $P_{0}$, but in ways that depend on the distribution and are
not easy to quantify. Optical images and color maps can be used to identify
and mask any dusty regions. Typically the SBF measurement is confined to
“clean regions,” and we assume all dust patches are identified and masked;
thus, no further contribution is considered in $P_{0}$.

In most cases, the globular clusters (GCs) within the target galaxy are the
main source of contamination for $P_{0}$, with faint background galaxies being
a lesser contaminant. We estimate the extra power from these sources by
fitting, down to the detection limit, a combined luminosity function that
includes a radially-dependent Gaussian GC luminosity function (GCLF; e.g.,
harris01 ) and a power-law LF for the galaxies. We then estimate the “residual
power” $P_{r}$ from sources fainter than this limit (which varies with radius)
by extrapolating the fitted combined LF, multiplied by the square of the
source flux, from the radially-dependent detection limit down to zero flux
(see blake95 ). This residual power is then subtracted from $P_{0}$ to obtain
the intrinsic stellar fluctuations, typically denoted $P_{f}=P_{0}-P_{r}$.

Figure 1: Illustration of SBF observations and measurements. $(a)$ Simulation
of the stellar population in a spheroidal galaxy at the distance of the Virgo
cluster ($D_{Virgo}\simeq 16.5$ Mpc blake09 ) as observed with the E-ELT in
$\sim$1 hour (Cantiello et al., 2023, in prep.). $(b)$ Same as in panel $(a)$,
but for a galaxy ten times more distant. $(c)$ Same as in panel $(a)$, but for
a galaxy fifty times more distant. Stars, which appear marginally resolved in
panel $(a)$, blend together into a smooth brightness profile at larger
distances. $(d)$ Near-infrared image of NGC 1399 from the HST WFC3 camera.
$(e)$ Model of NGC 1399’s surface brightness distribution derived from the
WFC3/IR image. $(f)$ Residual frame, obtained from the galaxy image $(d)$
minus the model $(e)$. $(g)$ Typical luminosity function analysis for
estimating the “residual variance” $P_{r}$ due to contaminating sources: green
squares show the data, the blue curve and red line show the fits to the
globular cluster and background galaxy luminosity functions, respectively, and
the solid black line is the combined model luminosity function (data and fits
are fromcantiello11 ). The vertical gray dashed line indicates the GCLF
turnover magnitude and the shaded area shows the magnitude interval where the
detection is incomplete. $(h)$ Color-magnitude diagram of an old stellar
population (data for the MW globular cluster NGC 1851 piotto02 ); the RGB/AGB
population is highlighted with red dots. $(f)$ A schematic illustration of the
SBF power spectrum analysis. Images reproduced with permission from moresco22
; piotto02 .

Because the GCs have a radial density distribution that increases toward the
galaxy center, while the detection limit also gets brighter near the center,
the fluctuations from GCs in the central region can overwhelm the signal from
the stellar fluctuations, especially for massive galaxies with rich GC
systems. Space-based observations (or high-resolution ground-based data
obtained with adaptive optics [AO]) can greatly improve the distance limit for
SBF by reducing the effect of crowding and pushing the detection limit fainter
for the GCs. The surface density of background galaxies is typically lower and
relatively uniform over the area, which means that their impact is less
dramatic and is easier to robustly constrain.

Finally, the SBF magnitude is then calculated as
$\overline{m}=-2.5\log(P_{0}-P_{r})+m_{\mathrm{ZP}}$, where $m_{\mathrm{ZP}}$
is the zero point magnitude. For a schematic illustration of the intricate SBF
analysis procedure, see Fig. 1.

###  0.3.2 SBF calibration

To convert $\overline{m}$ into a distance, knowledge of the absolute SBF
magnitude $\overline{M}$ of the population is required. The most common method
for estimating $\overline{M}$ is through empirical relationships. Studies of
large samples of galaxies at known distances or in compact groups have shown
that $\overline{M}$ can be described as a one-parameter family with integrated
color (e.g., $V{-}I$ or $g{-}z$) as a useful parameter. The ground-based SBF
survey of Tonry and collaborators tonry97 ; tonry00 ; blake99 ; tonry01 found
that a linear relation describes accurately the color dependence of
$\overline{m}_{I}$ in nearby groups and clusters. They also showed that
$\overline{M}$ in the $I$ band is a universal function of color, with an
intrinsic scatter in the range of 0.05 to 0.10 mag. Later studies confirmed a
similar intrinsic scatter in the calibration relation for red galaxies in
suitable passbands blake09 ; jensen15 , although for high-quality SBF
measurements over a large color range, the $\overline{M}$-vs-color relation is
more accurately described by a higher-order polynomial curve blake09 .

Since the introduction of SBF, many empirical calibrations of $\overline{M}$
as a function of various optical and near-IR colors have been presented bva01
; blake10 ; mei07 ; jensen15 . However, a major drawback of empirical
approaches is the need to establish a new calibration whenever the reference
passband for SBF or the reference color of the observing campaign changes. An
alternative approach is to use SBF magnitude versus color relations derived
from stellar population synthesis models bva01 ; cantiello03 ; worthey93 . The
advantage of theoretical calibrations is that $\overline{M}$ can easily be
computed for any passband and reference color. Furthermore, fully theoretical
calibrations are independent of other distance indicators, avoiding systematic
uncertainties associated with the primary indicator. Consequently,
theoretically calibrated SBF can be regarded as a primary (though not
geometric) distance indicator blake12b .

There is generally good agreement in SBF predictions from independent stellar
population synthesis models in red optical bands. Discrepancies arise when
comparing SBF magnitudes predicted by different models in bluer passbands or
in the near-IR, where the models are not yet as robust jensen15 . These
discrepancies can exceed $\sim$0.2 mag. Such a difference is considerably
larger than the empirically derived scatter, rendering stellar population
model predictions unreliable as an alternative to empirical $\overline{M}$.
Therefore, except for a few notable cases biscardi08 , numerically derived
$\overline{M}$ vs color relations serve mainly as validation tools for
empirical calibrations.

Before concluding this subsection, it is important to emphasize that accurate
distance estimation using $\overline{M}$ calibrations requires precise galaxy
colors. Hence, a reliable photometric calibration is crucial to ensure
consistency and precision in the distances derived.

###  0.3.3 Reference targets and passbands

Owing to the definition of SBF and the measurement procedures outlined above,
the ideal target galaxy for fluctuation measurements should have minimal dust,
a brightness profile that is sufficiently regular, and a relatively
homogeneous population of stars with a constant or at least not highly
variable stellar luminosity function across its isophotes. This ensures a
relatively constant SBF signal across the regions of the galaxy where the
measurement is carried out, reducing the intrinsic variation of $\overline{m}$
and resulting in a more accurate distance estimate with lower errors.

In short, the ideal SBF targets are passively evolving, massive red
ellipticals that are free from dust contamination and have a well-mixed
population of stars with no recent star formation. Empirical calibrations have
shown that such galaxies exhibit a tight correlation between absolute SBF
magnitude and optical or near-IR colors, enabling SBF distances with a
contribution from intrinsic scatter as low as 0.06 mag in certain
passbands222The final distance errors are larger due to measurement
uncertainties, as discussed in Sec. 0.4. blake09 ; blake10 ; carlsten19 .
Stellar population models also support the narrow scatter of $\overline{M}$
for colors typical of “red and dead” galaxies with stellar populations
dominated by an old, metal-rich component bva01 ; cantiello03 .

While the SBF method was originally developed for bright, morphologically
regular early-type galaxies, it has more recently been used for a wider
variety of stellar systems, including bulges of spirals and the low mass
ultra-diffuse galaxies blake18 ; cohen18 . In contrast to massive red
ellipticals, bluer galaxies with intermediate masses have properties
consistent with younger ages, lower metallicities, or both. Galaxies in this
color regime have more uncertainties from the age-metallicity degeneracy333The
age-metallicity degeneracy refers to the effect of stellar populations of a
specific age and metallicity displaying similar colors, SBF magnitudes, and
certain spectral indices as populations with a younger age and higher
metallicity, or vice versa worthey94 ., which can result in targets with
different stellar populations having the same integrated color. However,
having the same integrated colors (which are mostly dominated by main sequence
stars) does not always mean the same SBF absolute magnitude. This effect
implies a larger scatter in the $\overline{M}$-vs-color relations as compared
to red massive galaxies, leading to larger scatter associated with the derived
distances.

Additionally, galaxies with intermediate colors often host an intermediate-age
stellar population (3-7 Gyr), which includes many asymptotic giant branch
(AGB) stars. These stars are in a bright and rapidly evolving phase, hence
their stochastic appearance can make the SBF signal less stable, especially in
near-IR bands raimondo09 . Although AGBs have little impact on integrated
colors they represent an additional source of scatter to the SBF in these
galaxies. As a result of degeneracy effects and AGB stars, in some bands
intermediate-mass targets suffer a larger scatter in SBF amplitude at fixed
color, which makes it challenging to measure individual distances with an
accuracy similar to the red-massive galaxies.

The accuracy of the SBF technique further drops for bluer, low-mass, low
surface brightness galaxies. This is partly because of increased scatter from
stellar population effects, including stochastic variations in the AGB stars.
Additionally, measuring SBF in diffuse blue galaxies is challenging because
low stellar densities make it harder to achieve an adequate signal-to-noise
for the stellar-count fluctuations. As a result, a precision of approximately
15% should be considered the norm for such galaxies under optimal conditions
greco21 .

Despite these challenges, the development of large-format imagers and wide-
field surveys on 4m-class telescopes has increased the interest in SBF as a
tool for determining distances to smaller galaxies. Recent studies have
applied the SBF technique as a distance indicator for galaxies with surface
brightness levels as low as $\mu_{r}\sim 24$ mag/arcsec2 greco21 ; carlsten19
; kim21 ; blake18 . In these cases, the technique is typically not exploited
for its accuracy but rather because these faint and diffuse galaxies do not
offer many reliable alternatives for measuring distances. Despite the larger
uncertainty, the SBF technique represents a unique opportunity to confirm
galaxy associations and group or cluster membership for candidate satellite
galaxies.

Concerning the preferred passbands, SBF distances have been measured in
various filters, ranging from visible to near-infrared wavelengths. Shorter
wavelength bands are generally avoided because the SBF magnitudes are much
fainter and more susceptible to contamination from GCs and dust. Nevertheless,
SBF data in bluer bands can be valuable tracers for studying unresolved
stellar populations due to their sensitivity to stellar population properties
worthey93 ; bva01 ; cantiello03 ; rodriguez21 . Redder optical ($I$ and $z$)
and nearer IR ($YJH$) bands are instead preferred for distance measurements.

Empirical studies have shown that the intrinsic scatter of $\overline{M}$ at a
fixed galaxy color is around $\sim 0.06$ mag blake09 ; blake10 for the most
massive ellipticals in optical bands. In the near-IR bands, the intrinsic
dispersion of $\overline{M}$ is closer to 0.10 mag due to the larger variance
arising mainly from bright AGB stars jensen15 ; cantiello18 . Despite the
somewhat larger instrinsic scatter, near-IR bands such as $J$ and $H$ have
several advantages. The SBF is intrinsically brighter, up to three magnitudes
brighter in $K$ than $I$, although the intrinsic scatter in $K$ is significant
and not well characterized. Near-IR bands also offer a much more favorable
contrast with respect to GCs. Reliable $\overline{m}$ measurements need to
mask GCs to at least $\sim 0.5$ mag fainter than the GCLF peak in the $I$
band, while contamination can be reduced to the same level in the $K$ band by
reaching $\sim\,$2 mag brighter than the GCLF peak. Finally, the impact of
residual dust contamination in the near-IR is negligible in most cases
luppino93 .

The region around $\sim\,$1$\,\mu m$ ($Y$ band) is particularly interesting
for SBF. According to independent stellar population models, and as expected
from the inversion of the slope of the empirical SBF vs color relation between
optical and near-IR bands for intermediate-color evolved galaxies, this
wavelength should exhibit nearly complete degeneracy with respect to stellar
population: $\overline{m}$ remains constant with color, making it an ideal
distance indicator worthey93 ; cantiello03 . In practice, this does not hold
for the reddest, most massive ellipticals, for the which the SBF continues to
get fainter with color, but these galaxies are comparatively rare. For future
wide-area surveys the $\sim Y$ band may be the best choice for SBF
measurements, although further study is needed.

In summary, there not a definitive spectral “sweet spot” for measuring SBF,
and any passband from $\sim\,$0.8 to $\sim\,$2 $\mu m$ has its pros and cons.
With the rapid development of near-IR technologies and the significant
investment in future astronomical facilities in these passbands (e.g., Roman,
the ELTs, and the recently launched JWST), along with the brighter intrinsic
luminosity and negligible effect of dust contamination, it seems likely that
the future of the SBF method, in terms of distance/depth and number of targets
reached, lies primarily in the near-IR.

##  0.4 Uncertainties

###  0.4.1 Statistical uncertainties

Statistical uncertainties affecting SBF can be grouped into three main
categories: $\overline{m}$ measurement errors, uncertainty in $\overline{M}$
due to photometric errors in the galaxy color and intrinsic stellar-population
scatter in the calibration relation, and uncertainties from data processing
and calibration. The last category includes effects like flat fielding and
uncertainty in the photometric calibration. While not unique to SBF, it is
important to note their presence since they can impact various parts of the
SBF measurement process in non-trivial ways. The uncertainty in the flat-
fielding multiplicative factor affects the $\overline{m}$ measurement
affecting both the variance and average term in eq. 2 and the color used in
the calibration equation. Any sources of uncertainty in the reduction stage
impact in a relatively intertwined way the final uncertainty on the distances
from SBF. However, these uncertainties are typically well-controlled with
standard data reduction and calibration procedures, with a typical level of
$\sim 1\%$. Regarding SBF measurement uncertainties, these are directly tied
to the specific steps required for the measurement. Usually, each error term
introduced during any of the various analysis stages described in section
0.3.1, can be independently estimated and then combined in quadrature with the
others. The most relevant error terms are the following.

  * •

Sky background. The process of sky subtraction has an associated error due to
spatial and/or temporal variations in the sky emission (especially in the
near-IR, where the sky background varies on the time scale of the
observations), scattered light, and the extended nature of the galaxy,
especially if the field of view is comparable to the galaxy size. The effect
of the sky uncertainty increases significantly in the outer parts of the
galaxy and for galaxies of lower surface brightness. To determine the final
error associated to the sky uncertainty, the entire SBF measurement may be
repeated by offsetting the background by the uncertainty level; this alters
the final $\overline{m}$, and the size of the change is then taken into
account for the final uncertainty. The sky uncertainty typically contributes
$\sim 0.02$ mag of the total error on $\overline{m}$ in the near-IR, and less
than that in optical bands;

  * •

Point Spread Function. Accurate characterization of the PSF is essential for a
reliable SBF measurement, and this depends on the number of bright,
unsaturated stars in the field. Thus, for a given galaxy, the quality of the
PSF template depends on the size of the field, stability of the PSF across the
field, signal-to-noise, saturation level, etc. Once a reliable set of
reference stars is identified, the SBF measurement is repeated using each
available PSF and possibly a composite PSF model. The standard deviation of
these measurements, typically less than $\sim 0.03$ mag, is then used as the
PSF-fitting uncertainty;

  * •

Power Spectrum Fitting. The accuracy of the power spectrum fit is affected by
several factors, such as the number of good pixels in the annulus, the spatial
structure of the mask, and the presence of patterns in the residual image
caused by bars, shells, tidal interaction features or spiral arms. Low- and
high-wavenumber power excesses or deficits can be filtered out during the
power spectrum fit, yet the range of useful $k$-numbers needs to be large
enough for reliable fitting. The fitting of the PSF power spectrum to the
galaxy-subtracted data involves a statistical fit uncertainty of $\sim$0.02
mag.

  * •

Residual Variance $P_{r}$. Various approaches have been used to determine the
uncertainty in correcting for variance from GCs and background galaxies that
are too faint for direct detection. One approach is to examine the change in
$P_{r}$ for different LF parameters and source detection thresholds. Another
involves using different, independent detection and photometry tools
(SExtractor, DAOphot, DoPHOT, etc.). Based on a variety of tests, a typical
uncertainty of about 25% is adopted for the $P_{r}$ estimate; thus, to ensure
the contribution to the total error on $\overline{m}$ from the $P_{r}$
correction is $<0.05$ mag requires that $P_{r}/P_{f}\sim 0.2$. To keep this
error term at an acceptable level, it is essential for the images to have
sufficient depth; this is vastly easier to achieve with the superior
resolution and reduced background of space-based observations.

  * •

Extinction. Another component in the error budget is the uncertainty in the
extinction correction, typically about 10% of the extinction correction
itself. This is added in quadrature to the other sources of statistical error.

As for uncertainty in the adopted value of $\overline{M}$ for a given galaxy,
apart from the systematic zero-point error from the distance calibration
discussed in the next section, the statistical error in $\overline{M}$
includes terms arising from the error in the integrated color of the galaxy
(needed to estimate $\overline{M}$ from the calibration relation) and from the
intrinsic, irreducible scatter in the calibration equation. The latter effect
results from stellar population variations and varies with the bandpass.

Overall, SBF measurement errors for well designed observations can achieve
statistical uncertainties of $\sim\,$0.05 mag in $\overline{m}$. For a red
galaxy observed in a filter close to 1 $\mu$m, the intrinsic scatter in
$\overline{M}$ is $\sim\,$0.06 mag. Hence, when combined, these errors result
in a purely statistical uncertainty that can be as low as 0.08 mag, or 4% on
the distance. Nonetheless, fainter, blue galaxies and sub-optimal observing
conditions will have larger statistical uncertainties.

###  0.4.2 Systematic uncertainties

The main systematic uncertainty in SBF distances is in the $\overline{M}$ zero
point444As future facilities become capable of reaching distances well beyond
100 Mpc, SBF $k$-corrections that take the galaxy spectrum redshift properly
into account may be necessary to avoid introducing further systematic effects
jensen21 . As of now, the 100 Mpc limit is usually not surpassed, so the
correction does not pose any problems when addressing systematic
uncertainties.. Empirical calibrations of $\overline{M}$ are typically based
on the zero point from the SBF ground-based survey by Tonry and colleagues,
which obtained SBF for bulges of spiral galaxies with distance estimates from
Cepheids tonry01 . A recent reassessment of this calibration reports a total
systematic uncertainty of 0.09 mag in the Cepheid-based SBF distance zero
point blake21 . This includes contributions from the tie between near-IR and
optical SBF distances, the tie between SBF and Cepheids, and the current
uncertainty in the Cepheid zero point, taking into account the improved
distance to the Large Magellanic Cloud from detached eclipsing binaries
pietrzynski19 .

The largest source of systematic uncertainty in this calibration is the tie
between SBF and Cepheids. This can be improved to a limited degree with
further refinements in Cepheid luminosities with Gaia parallaxes (e.g.,
clementini19 ) and by using more modern Cepheid distances, including a
recently revised distance to M31 (one of the SBF calibrators) with a precision
better than 2% li21 . New space-based SBF measurements for all the calibrator
galaxies would also help. However, it is important to note that the ideal
galaxies for measuring SBF (red ellipticals, Sec. 0.3.3), will never have
Cepheid distances because they have no recent star formation.

An alternative to calibrating SBF from Cepheids is to use the tip of the red
giant branch (TRGB) method. The TRGB method is ideal for measuring distances
to early-type galaxies and can be calibrated using geometric distances from
Gaia (e.g., dixon23 ; li23 ). Unlike the case for Cepheids, the stellar
populations underlying both the SBF and TRGB methods are the same, namely old
low-mass stars. Given this, TRGB distances are the natural choice for
calibrating SBF. In an early effort to calibrate SBF using TRGB, mould09
examined a set of 16 galaxies within 10 Mpc. These were predominantly blue,
low-mass early-type galaxies, and thus the scatter was large, although the
zero point was consistent with the Cepheid calibration.

A high-quality TRGB-based SBF calibration must be based on massive red early-
type galaxies. Recently, blake21 compared SBF and TRGB distances to the few
such galaxies for which both types of measurment exist, including the bright
ellipticals M 60 and M 87 in Virgo, and the dusty merger remnant NGC 1316 in
Fornax. The authors found that the mean offset between the TRGB and Cepheid-
calibrated SBF distances of the two galaxies in Virgo was only ${-}0.01$ mag.
NGC 1316, a far less ideal galaxy, had a larger offset of 0.14 mag. Although
preliminary, this result shows no significant difference between Cepheid- and
TRGB-based SBF calibrations.

With the launch of JWST, the prospects have never been better for improving
the SBF zero-point calibration using only modest amounts of observing time. A
recently approved Cycle 2 program will measure TRGB distances and SBF
magnitudes for 14 luminous, red early-type galaxies reaching out to 20 Mpc.
The sample is designed to produce an absolute calibration for the SBF method
of better than 2%. Further reduction in the zero-point uncertainty can be
achieved by combining the TRGB- and Cepheid-based calibrations.

In summary, the current systematic uncertainty on SBF distances using the
Cepheid-based calibration is 0.09 mag or about 4.2%. This value could be
improved by obtaining new SBF measurements for the bulges of spiral galaxies
that host Cepheids, but this approach will never be ideal because the best
targets for the SBF method are massive red ellipticals. The preferred route
for substantially reducing the systematic uncertainty in $\overline{M}$ is via
the the TRGB method, which relies on the same underlying stellar population as
the SBF method. A new JWST program, measuring both the TRGB and SBF for a
well-selected set of early-type calibrators, should soon decrease the
systematic uncertainty in SBF distances to below 2%. Further improvements can
be achieve with additional TRGB calibrators and/or combining the TRGB and
Cepheid calibrations.

##  0.5 Hubble-Lemaitre Constant

###  0.5.1 Current constraints from SBF

Since its introduction for measuring extragalactic distances, the SBF method
has been applied to about 400 galaxies. The random uncertainty associated with
these distances varies greatly, from $\sim\,$4% for space-based observations
of massive, red galaxies, to $\sim\,$20% for dwarf galaxies observed from the
ground. The distances obtained have been used for a variety of scientific
cases. For example, SBF has been used to measure distances to:

  * •

NGC 4993, the lenticular galaxy host of the gravitational wave event GW 170817
which had the first observed electromagnetic counterpart cantiello18 ;

  * •

M 87, the cD host galaxy of the first supermassive black hole to have its
“shadow” imaged using the EHT ehtVI ;

  * •

hundreds of other member galaxies in the Virgo and Fornax clusters to explore
the substructure in these environments blake09 ; cantiello18b ; dunn06 ; mei07
;

  * •

numerous ultra-diffuse galaxies cohen18 , including NGC 1052-DF2, the galaxy
proposed to be devoid of dark matter blake18 ; vdk18 .

In addition, as the method reaches into the Hubble flow, many authors have
used this technique for the estimation of the Hubble constant.

The first study to estimate $H_{0}$ directly from SBF distances decisively in
the Hubble flow used a sample of 15 galaxies from 40 to 130 Mpc observed with
NICMOS on HST and obtained $H_{0}\approx 76$ km s-1 Mpc-1 (dropping to
$\sim\,$72 when restricted to the six most distant targets).555These numbers
should be updated to the most recent Cepheids calibration, which has a minor
impact on the final $H_{0}$ value. Another early direct measurement of $H_{0}$
from SBF used archival HST/ACS observations combined with a theoretical
calibration based on population models and likewise found $H_{0}=76$ km s-1
Mpc-1 biscardi08 .

Among recent results, we mention three studies that are closely related to
each other. The first study, by khetan21 (hereafter K21), presented a re-
calibration of the peak magnitude for a sample of 24 local Type Ia supernovae
(SNe Ia) using SBF distances and a hierarchical Bayesian approach. The authors
then extended the calibration to a sample of 96 SNeIa at redshifts
$0.02<z<0.08$ and obtained $H_{0}=71.2\pm 2.4\,(\rm{stat})\pm 3.4\,(\rm{sys})$
km s-1 Mpc-1, where we have updated the value to use the improved LMC distance
from pietrzynski19 for consistency with the other recent $H_{0}$ values
discussed here. K21 used a diverse sample of SBF distances, which were
collected from observations over a span of twenty years, along with an equally
diverse sample of SNeIa. While normalizing the SBF distance sample to a common
reference calibration was difficult, the heterogeneity of the sample provided
some benefits, as it should in principle be less prone to systematic effects,
such as those related to specific instruments or data analysis procedures.
However, this also resulted in large final statistical uncertainties. The
value of $H_{0}$ reported by K21 was roughly midway between those found
directly from Cepheid-calibrated SNe Ia riess19 and predicted from analysis of
the cosmic microwave background (CMB) planck20 .

In another recent study, Blakeslee et al. blake21 (hereafter B21) used the
catalog of SBF distances from Jensen et al. jensen21 to estimate $H_{0}$. This
sample comprises 63 bright early-type galaxies observed with the WFC3/IR
instrument on the HST, reaching distances up to 100 Mpc. The resulting value
of $H_{0}=73.3\pm 0.7\,\mathrm{(stat)}\pm 2.4\,\mathrm{(sys)}$ km s-1 Mpc-1
agrees well with most other direct measurements in the local universe, but
deviates significantly (by about 2.5 $\sigma$, including the systematic
uncertainty) from the CMB prediction, assuming the standard $\Lambda$CDM
model. It is noteworthy that the random uncertainty is about one third of that
reported in K21.

To ensure the reliability of their findings, B21 conducted an analysis of
$H_{0}$ involving four different treatments of galaxy velocities, including
group-averaged CMB frame velocities, individual CMB velocities, and flow-
corrected velocities from two different models. Additionally, their final
$H_{0}$ combines the mutually consistent SBF calibrations from Cepheids and
the TRGB in order to reduce the final systematic error down to the
$\sim\,$3.3% level. As noted previously, the number of TRGB calibrators in
this analysis was very small, and future studies incorporating a larger sample
of giant ellipticals with TRGB distances measured by JWST and tied to Gaia
parallaxes should vastly reduce this systematic error.

The third recent work garnavich22 related to $H_{0}$ combines an approach like
that of K21 with the WFC3/IR SBF distances from jensen21 . The authors of
garnavich22 used SBF distances for 25 host galaxies of SNe Ia to provide an
absolute calibration for the Pantheon+ compilation scolnic22 of 1500 SNe Ia
extending far out into the Hubble flow. Using the original Pantheon+ “SALT2”
parameters for the dependence of peak luminosity on decline rate and color,
these authors find $H_{0}=74.6\pm 0.9\,\mathrm{(stat)}\pm
2.7\,\mathrm{(syst)}$ km s-1 Mpc-1, which agrees to within ${\sim\,}{1}\sigma$
with the value obtained by B21 (the systematic error is common between the two
studies, so the comparison must be done with respect to the statistical error
only).

However, it is important to note that the SNe Ia whose luminosities can be
calibrated with SBF distances are generally fast decliners in massive host
galaxies. When garnavich22 limit their analysis to fast-declining SNe Ia in
massive hosts and rederive the SALT2 parameters, the SBF-calibrated $H_{0}$
from these SNe becomes $73.3\pm 1.0\pm 2.7$ km s-1 Mpc-1, which is identical
to the value derived by B21 from SBF distances in the Hubble flow. This
$\sim\,$1$\sigma$ change in $H_{0}$ suggests a slight difference between the
best-fit SALT2 parameters for the relatively small set of fast-declining SNe
Ia in massive galaxies as compared to the full Pantheon+ sample.

###  0.5.2 Forecasts for new and future facilities

As outlined in the previous section, recent SBF-based estimates of $H_{0}$
align more closely with the results found by Cepheid-calibrated SNe Ia than
with the predictions from the CMB. The potential use of SBF, particularly with
JWST and upcoming astronomical facilities, offers a promising route to derive
a more precise and accurate value of $H_{0}$ that is fully independent of SNe
Ia and Cepheids. Here, we present the expected results of the SBF method for
estimating $H_{0}$ using various existing and forthcoming facilities: JWST,
Rubin Observatory, ESA’s Euclid mission, the Roman Space Telescope, and the
Extremely Large Telescopes (ELTs).

To evaluate the potential for constraining $H_{0}$ from SBF with each new or
forthcoming facility, we proceed as follows. First, we estimate the maximum
distance $d_{Max}$ that can be reached with the specific facility, as
explained below.

Then, we count the number of massive elliptical galaxies that can be observed
by the telescope/instrument within a distance range of $d_{min}\leq D\leq
d_{Max}$. Here, we assume $d_{min}=40$ Mpc to exclude galaxies in the local
universe where flow motions and peculiar velocities are large compared to
distance errors. To estimate $d_{Max}$, we assume (as is generally the case)
that contamination by GCs is the limiting factor for well-designed SBF
measurements. Thus, $d_{Max}$ is the distance to which the GCs can be detected
and removed to a faint enough limit, and the residual contamination reliably
estimated, so that the error in the correction drops below the intrinsic
scatter in the method. For instance, in the $I$ band, where the peak of the GC
luminosity function (GCLF) is at $M_{I}\approx-8.0$ AB mag, accurate SBF
measurements require detecting and removing sources to +0.5 mag fainter than
the GCLF peak, or $M_{I}\approx-7.5$ AB mag. In the $H$-band (which has
slightly larger intrinsic scatter), the same relative amount of contamination
from GCs can be reached for a detection limit about 1.5 mag brighter than the
GCLF peak; jensen15 ). We estimate the near-IR GCLF peak magnitudes from
nantais06 and convert to the AB system.

The expected depth is determined based on available information for the
specific facilities. For instance, in the case of Rubin Observatory’s LSST, we
use the expected 10-year $5\sigma$ point-source depth for the $i$ band, with a
GCLF peak of $\sim-8.0$ AB mag and sky coordinates with
$\hbox{Dec}(\hbox{\small J2000})\leq 15^{\circ}$.

Another important factor is the selection of an appropriate list of targets.
We use the 2MASS Redshift Survey (2MRS), which comprises approximately 43,000
galaxies brighter than $K_{s}$ = 11.75 mag (Vega) huchra12 . This survey is
nearly complete, with the exception of regions near the Galactic plane, which
account for only 9% of the sky, and an $L^{*}$ galaxy is included if it is
within $\sim$135 Mpc. We first focus on “optimal” targets, i.e., bright
galaxies with a morphological $T$-type $\leq-4$ (indicating morphologically
regular ellipticals) and an absolute $K$-band Vega magnitude $M_{K}\leq-25$
mag (similar to M87), where distance is estimated from the 2MRS redshift
assuming $H_{0}=75$ km s-1 Mpc-1. Under these assumptions, Table 1 (column 7)
reports the numbers of optimal targets reachable by each of the new and
upcoming facilities mentioned above.

Now, if we relax the selection somewhat on the galaxy morphology and
luminosity, accepting $T\leq-1$ and $M_{K}\leq-22.5$ mag as “good” candidates,
then the numbers of SBF targets increase dramatically, as again shown in
column 7 of the table. It is important to note that these numbers represent
conservative lower limits. For reference, the NGVS survey on the CFHT, with a
median $i$-band seeing of $0\mbox{$.\\!\\!^{\prime\prime}$}7$ and 5$\sigma$
depth three magnitudes fainter than the GCLF peak, enabled SBF measurements
for about 300 galaxies in the Virgo cluster cantiello18b down to
$M_{K}\sim-16$ Vega mag. It was also possible to measure SBF in many galaxies
with positive T-type, such as bulges of spirals and galaxies with some dust
contamination, thanks to the availability of $u$-band data. Thus, the numbers
of candidates measurable with the facilities specified in Table 1 could
increase by a factor of a few, or even orders of magnitude, when fainter and
more irregular targets are considered.

For estimating the constraints on $H_{0}$, we also need to adopt values for
the expected distance errors. Thus, for ground-based observations in natural
seeing (Rubin/LSST), we conservatively assume a total error in the SBF
distance of 8% for the optimal target galaxies, and 16% for the “good”
targets. For high-resolution space-based and AO-assisted observations, we
assume total errors of 4% and 8% for the optimal and “good” targets,
respectively.

All the necessary parameters required for estimating the number of reachable
targets with the selected facilities are listed in Table 1, along with the
estimated maximum distances and sample sizes for each facility in selected
bands. Taking Rubin as an example, it is expected that approximately 50
optimal targets within $d_{Max}\approx 70$ Mpc will be measurable in the
Baseline survey area, resulting in a constraint on $H_{0}$ of $\sim 1.2\%$
(stat). For “good” targets, the estimated number increases to 1500, and the
statistical error drops well below the systematic uncertainties from the
distance calibration and possible large-scale flow motions.

The Roman Space Telescope is especially promising for future constraints on
$H_{0}$ using SBF because of its combination of HST-like resolution, excellent
PSF stability, near-IR coverage, and planned large-area surveys blake23 .
While awaiting the Roman telescope, however, the JWST has unsurpassed
capabilities for targeting galaxies one at a time out to $\sim\,$300 Mpc, an
unprecedented distance for SBF that exceeds the limit of claimed “Hubble
bubbles” (e.g., keenan13 ; romano18 ). In the NIRCam passbands reported in
Table 1, we find that around a thousand candidates from the 2MRS catalog will
be optimal targets for JWST. Of course, only a few dozen are needed to bring
the statistical error comfortably below 1%, so there will be no shortage of
targets from which to choose.

To demonstrate the feasibility of SBF measurements with JWST/NIRCam, we
conducted a preliminary analysis on NGC 7317, the elliptical galaxy in
Stephan’s Quintet at a distance of about 100 Mpc, using the public Early
Release Data.

The characteristic SBF signal is clearly observed across all passbands,
including mid-infrared bands that have not previously been used in SBF
studies. The left panels of Figure 2 display the measured power spectra in the
F150W band of the short-wavelength channel and the F277W band of the long-
wavelength channel. The other panels show the ratio of the fluctuation
amplitudes on- and off- the galaxy center and the $P_{0}$ coefficient of the
PSF term in the power spectrum, using a star within the image as the reference
PSF. Analysis of the SBF signal for this target is in progress, and further
work is needed to improve the agreement with the PSF templates, but the signal
is very strong, given the short exposure times for this distance.

With the forthcoming generation of wide-area imaging surveys such as
Rubin/LSST, Euclid, and the Roman Community Surveys providing hundreds or even
thousands of galaxies suitable for SBF measurements, it is essential that the
SBF analysis should be carried out in a robust, streamlined, and automatic
way. Historically, the limited number of available SBF measurement pipelines
have been poorly automated and required extensive user intervention. This is
especially true for the galaxy fitting and residual masking, as these steps
must be optimized to ensure clean power spectra. However, it is clear that the
method needs to be adapted to run in “production mode” to deliver robust
results with minimal human intervention. Multiple parallel efforts are ongoing
in this regard.

The prospects for SBF measurements with AO-assisted observations on 30-40m
telescopes are also exciting. Table 1 reports the expectations for
E-ELT/MICADO with the MORFEO AO module. With their extremely large apertures
and diffraction-limited resolution $<0\mbox{$.\\!\\!^{\prime\prime}$}02$, such
facilities should be able to measure SBF distances out to redshift $z\sim
0.1$. However, measuring SBF with AO-assisted instruments may be challenging
because of the spatially and temporally varying PSF, significant overheads,
and the requirement for accurate $k$-corrections at these distances jensen21 .
Nevertheless, once the technique is optimized for 30m-class telescopes, a
carefully selected set of targets could have a significant impact, potentially
providing an independent direct detection of cosmic acceleration.

Figure 2: Power spectrum analyses of JWST data on NGC 7317, an elliptical
galaxy at ${\sim\,}$100 Mpc, in the F150W (top row) and F277W (bottom row)
passbands. Left panels: power spectra of the central masked region after
galaxy subtraction (blue solid curve), the scaled PSF (red curve), and an
empty region away from the galaxy (orange curve); dotted blue lines show the
range of the starting waveneumbers $k$ for the fits. Middle panels: ratio of
the power spectra on- and off-galaxy; the structure in this ratio for F150W
may indicate a variable PSF in the combined image. Right: Uncorrected power
spectrum normalization $P_{0}$ derived from the fits in the left panels as a
function of the starting $k$ value (the fitted $P_{0}$ values here are
insensitive to the starting $k$).

##  0.6 Conclusions

Over the last 30 years, the SBF method has become one of the most powerful
methods for measuring galaxy distances from the Local Group to the Hubble flow
and constraining the value of $H_{0}$. Unlike other precision distance
indicators (Cepheids, TRGB, SN Ia, masers), SBF measurements require only a
modest investment of telescope time, especially when using high-resolution
space-based data. SBF distances to large numbers of galaxies observed with HST
using WFC3/IR have already constrained $H_{0}$ to 3.4%, where the main source
of uncertainty is the zero-point calibration. Now, with JWST/NIRCam, it
becomes possible to directly calibrate SBF using TRGB distances to the same
galaxies, opening a path to reduce the uncertainty on $H_{0}$ from SBF to
below 2%. The large aperture, superior resolution, PSF stability, and near-
infrared capability of JWST also make it possible to measure robust SBF
distances out to $\sim\,$300 Mpc. This will reduce the uncertainty from
possible large-scale bulk motions and vastly increase the number of optimal
targets in range of the method. Other forthcoming facilities such as Euclid,
Roman, Rubin, and the ELTs also hold great potential for the method. With an
improved TRGB-based calibration and samples of hundreds, or even thousands, of
SBF distances reaching out to hundreds of Mpc, a fully independent measurement
of $H_{0}$ to a precision approaching 1% will be in reach by the end of he
current decade.

Table 1: Maximum Distances and Numbers of Candidates for SBF Analysis Telescope | Band |  Depth(a) | GCLF | GCLF | $d_{\mathrm{Max}}$ | Optimal/good |  $\delta H_{0}/H_{0}$ (%)
---|---|---|---|---|---|---|---
(constraint) |  | (mag) | peak(b) | offset(c) | (Mpc) | candidates | optimal/good
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
Rubin (Dec$\leq 15^{\circ}$) | $i$ | 26.8 |  $-$8.5 |  $+$0.5 | 70 | 50/1500 | 1.2/$<$0.5
Euclid ($|b|\geq 23^{\circ}$) | $H_{Euclid}$ | 24.0 |  $-$8.6 |  $-$1.5 | 55 | 20/600 | 1.0/$<$0.5
Roman(e) | $J_{F129}$ | 28.0 |  $-$8.3 |  $-$1.0 | 230 | 1500/$>$5000 | 0.7/0.5(g)
Roman(e) | $H_{F158}$ | 28.0 |  $-$8.6 |  $-$1.5 | 300 | 1600/$>$5000 | 0.7/0.5(g)
JWST(e) | $J_{F115W}$ | 27.5 |  $-$8.3 |  $-$1.0 | 180 | 900/$>$5000 | 0.9/0.5(g)
JWST(e) | $H_{F150W}$ | 27.7 |  $-$8.6 |  $-$1.5 | 290 | 1600/$>$5000 | 0.7/0.5(g)
JWST(e) | $K_{F200W}$ | 27.8 |  $-$8.1 |  $-$2.0 | 310 | 1600/$>$5000 | 0.7/0.5(g)
ELT(f) (Dec$\leq 0^{\circ}$) | $K$ | 28.5 |  $-$8.1 |  $-$2.0 | 400 | 750/$>$4000 | 1.0/0.7(g)
$(a)$ Nominal $5\sigma$ point-source depth accounting for host galaxy
background.
$(b)$ Peak, or turnover, magnitude (AB) of the GCLF in each band.
$(c)$ Offset of the required detection limit with respect to the GCLF peak
(see text).
$(d)$ Estimated number of optimal/suitable SBF targets based on 2MRS huchra12
.
$(e)$ $5\sigma$ depth in one hour integration time.
$(f)$ Depth derived from the preliminary ELT/MICADO exposure time calculator,
assuming a $S/N=5$ in one hour in multi-conjugate adaptive optics mode.
$(g)$ Assuming that a fraction of $\sim 2\%$ of possible candidates are indeed
observed.

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