Stellar Candles for the Extragalactic Distance Scale

299 Pages·2003·14.866 MB·English

by  Barry F. Madore, Wendy L. Freedman (auth.), Danielle Alloin, Wolfgang Gieren (eds.)

1 The HST Key Project
on the Extragalactic Distance Scale:
A Search for Three Numbers
Barry F. Madore and Wendy L. Freedman
Observatories of the Carnegie Institution of Washington
813 Santa Barbara Street, Pasadena, CA 91101, USA
Abstract. We present a review of the main results of the Hubble Space Telescope
Key Project on the Extragalactic Distance Scale with emphasis on the new techni-
ques that were developed in order to undertake the observations, and the methods
thatwereadoptedinbothreportingtheresultsandquantifyingespeciallytheassocia-
ted statistical and systematic errors. The three numbers (the cosmological expansion
rate H , its statistical error and the systematic uncertainty, respectively) are H = 72
o o
±(3) ±[7] km/sec/Mpc.
1.1 Introduction
The Hubble Space Telescope (HST) was designed and built to measure the ex-
pansionrateoftheUniverse.Anditsucceeded.FordecadesbeforeHSTadebate
wasragingoverafactor-of-twodisagreementinthevalueoftheHubbleconstant.
After 8 years of observing and data reduction the final results of the HST Key
Project on the Extragalactic Distance Scale were published in Freedman et al.
(2001)[5].Precedingthatoverthirtypaperswerepublisheddetailingtheresults
on the discovery and measurement of Cepheids in individual galaxies constitu-
ting the Key Project sample. And with those publications the debate over the
Hubbleconstanthasbeenreducedtoandfocuseduponaninterestingdiscussion
of dominant systematics and residual random errors, but now at the 10% level.
TheoverallgoaloftheKeyProjectwastomeasuretheexpansionrateofthe
Universe, using Cepheids to calibrate a variety of independent, secondary di-
stance indicators so as to then reach beyond the locally perturbed flow out into
thecosmologicallydominatedexpansion.Giventhehistoryofhavingsystematic
errorsdominatingtheaccuracyofdistancemeasurements,theapproachadopted
by the Key Project was to explicitly assess systematics in any one approach by
examiningandusingseveraldifferentmethodsenroutetoaglobalmeasurement.
These secondary methods included surface brightness fluctuations, Type II su-
pernovae,theTully-Fisherrelationforspiralgalaxies,thefundamentalplanefor
elliptical galaxies, and finally Type Ia supernovae at the very farthest extreme
in distance. Each of the secondary distance indicators had their own strengths
and their own weaknesses; many overlapped in distance; some could be applied
to the same galaxies; all had their own systematics, both a cluster environment
and the field were being sampled. If systematic differences were to be found this
experiment was designed to highlight and to quantify them.
B.F. Madore and W.L. Freedman, The HST Key Project on the Extragalactic Distance Scale: A
SearchforThreeNumbers,Lect.NotesPhys.635,1–20(2003)
http://www.springerlink.com/ (cid:1)c Springer-VerlagBerlinHeidelberg2003
2 B.F. Madore and W.L. Freedman
Fig.1.1. Averaging over Systematics – An example of how averaging can reduce
the noise in the statistical determination of a standard metric. In this case the mea-
surement being undertaken is the length of a “standard rod” as defined by twenty
randomly selected “feet”
And so as the Cepheid observations were made and distances began to be
compiled the first order of business was to establish rigorous standards of docu-
menting,propagatingandreportingthestatisticalerrorsassociatedwithsample
sizes. This was paralleled by the enumeration and assessment of the various sy-
stematicerrorsassociatedwitheachdecisionandeverysteptakenalongtheway
toevaluatingdistancesandvelocitiescontributingtoafinalvalueoftheHubble
constant.
This review will not so much dwell on the value of the Hubble constant that
was finally reported, but rather the emphasis will be on the errors associated
with that determination.
1.2 “Statistics, Damned Statistics, and...”
Benjamin Disraeli is reported to have said that there are three kinds of lies:
“Lies, Damned Lies, and Statistics”. Curiously, it is now regarded that Disra-
eli never uttered these words and that Mark Twain, who in his autobiography
1 The HST Key Project on the Extragalactic Distance Scale 3
attributed this now famous phrase to the then Prime Minister of England was
himselfapparentlyindulginginalieofthefirstkind.Whethertherewasabitof
hiddenironythereonewillneverknow,butwhatisclearlymeantbytheoriginal
statement, whoever really said it, it is that at least in some contexts statistics
fall at the very bottom of the credibility heap. But that unfortunately, is where
any new study of the expansion rate of the Universe had to start. But in the
end, it was the careful application of statistics and error analysis that brought
the results of this study into sharp and final focus.
Theremaybethreetypesoflies,butforourpurposesthereareonlytwotypes
of errors: statistical errors and systematic errors. Statistical (or random) errors
usuallyareamenabletoreductionbyincreasingthesamplesize,N.√Theyobeya
random-walkconvergencearoundtheirmean,slowlydecreasingas1 N.Various
statistical errors can be combined in quadrature (if they are statistically (sic)
independent),andinsuchacaseafinallyreported,single,combinederrormakes
both mathematical and physical sense. Statistical errors measure precision.
Systematic errors are of an entirely different breed. They measure accuracy.
No matter how many times an experiment may be repeated, and no matter
how many samples may be taken, if the methodology is unchanged any inhe-
rent systematic errors will remain the same. Systematic errors are offsets and
displacements of the answer from the truth that no increase in sample size can
reveal or reduce. As such systematic errors are hard to evaluate even when they
are identified, and it is even harder to know when and whether they have even
been identified at all. After the noise of small numbers has been beaten down
by “statistically significant” sample sizes, one is always left “dominated by the
systematics”.TheHubbleconstantlongsufferedfrombothlarge,butunknown,
systematics, and from small, but over-worked, samples.
TheHubbleSpaceTelescopetookcareofthesamplesize,asshownbyTable3
in Freedman et al. (2001) [5] which lists the revised Cepheid distances to thirty-
one galaxies after the Key Project was completed. This is to be compared to
the handful of distances available in 1990, say, after more than half a century of
observing from the ground (see Madore & Freedman 1991 [7] for a compilation
of Cepheid distances relevant to that pre-HST period in time).
Dealingwithsystematicerrorsrequiredasoberenumerationoftime-honored
methods and an explicit evaluation of a host of implicit assumptions.
But HST did not just guarantee time to do the project as one would have
done it from the ground. It demanded that the whole project be optimized. We
had to know just how many observations of how many Cepheids in how many
galaxies would get sufficiently reduce the statistical errors. And then we had
to select those galaxies to cover and include as many tests for systematics as
we could conceive of. And do this before the shutter ever opened on the first
target. Below we outline how we optimized the scheduling of HST to monitor
a large number of galaxies to find significant samples of Cepheids, and how
the numbers of observations in the two filters were chosen so as to allow us to
discover variables, select out the Cepheids, measure their periods, amplitudes,
mean magnitudes and time-averaged colors, and accomplish all of this within a
4 B.F. Madore and W.L. Freedman
narrowwindowoftimeoftennotevenaslongastheperiodsofthelongest-period
Cepheids themselves.
1.3 Optimal Search Strategies
From the ground, after time is assigned on a given telescope, there is little
control one can exercise over the rising and setting of the Moon, or the motion
of weather patterns across the surface of the Earth. Even in sunny California
one can observe only when it is dark, and even then only when it is clear. The
situation from Earth-orbit is, of course, not the same; in many different ways.
Every ninety minutes there is an opportunity to open the shutter and begin
observing. Within those orbital constraints one can in principle schedule the
telescope to point at anything that is not occulted by the Earth, Moon or Sun.
HST could be scheduled in just that way. The challenge was to capitalize on
this feature and optimize the use of the telescope in monitoring a fair sample of
objects.
The historical precedent was not good. From the ground, Hubble, Baade,
Sandage and others spent decades semi-systematically (but still more or less
randomly) observing some of the most nearby galaxies (M31, IC 1613, NGC
6822, etc.) in search of variable stars. And with time and with great patience
results did flow in. Nevertheless the lunar cycle still imposed aliases on the
observations (Fig. 1.2), as extensive as they were, leaving gaps and clumps in
the phase-folded lightcurves of some of the variables. The situation was clearly
notoptimal.Infact,itwasnotevenpossibletoconsideroptimizingthesituation,
so little attention was paid to producing an observing strategy matched to the
problem; that is, not until HST came along.
WithHSTitwasnotonlypossiblebutitquicklybecamemandatorythatthe
telescope be scheduled in a highly optimized way. Time was extremely valuable
and competition was fierce. Furthermore it was not just one or two additional
galaxiesthatneededCepheiddistancesdetermineditwasanorderofmagnitude
more than had already been done from the ground that was required in order
to calibrate a variety of secondary distance indicators. So the challenge was:
Observe about a dozen different galaxies. Reduce the numbers of observations
from more than 100 per target down to order 10 epochs. Do this in two colors
(to measure reddenings). Detect the variables. Measure their periods. Extract
time-averaged luminosities, amplitudes and colors. And complete each set of
discovery/detection/measurementobservationsinasinglewindow,generallynot
exceeding 60 days.
Exposures had to be long enough to get good photon statistics on the indi-
vidual measurements of the stars, so as to unequivocally discriminate variables
from constant stars. Those same individual phase points had to be sufficiently
highinsignal-to-noisetoallowadelineationofthelightcurvesothatphasingof
thedataandaperioddeterminationcouldbemadethatwasinitselfsufficiently
precise that a robust period-luminosity relation could be constructed and false
non-Cepheids discriminated against either by the shape of their light curve or
1 The HST Key Project on the Extragalactic Distance Scale 5
Fig.1.2. A Sample of Light Curves of Cepheids in M31 as published by Baade &
Swope (1965) [1]. Note that even with over 70 observations, taken over a period of six
years there are still strong resonances in the data, especially for variables V326 and
V254whoseperiodsareveryclosetoanintegralnumberofdays,andinfactveryclose
to being exactly a week
by their colors, magnitudes and/or corresponding periods. Color observations
had to be woven into the observing schedule so that two independent apparent
moduli could be derived and reddening corrections extracted and applied to the
determination of a final true distance modulus.
Randomsamplingofanysourc√e(intrinsicallyvariableornot)willonlydrive
down the error on the mean as 1 N; but for a variable with an intrinsic am-
plitude larger than the observing errors the empirical demands are considerably
worse. In our case (for Cepheids with amplitudes expected to be anywhere up
to 1.5 magnitudes in the visual) the dominant error on the mean magnitude
and color is driven by the phase sampling of the light curve itself. Obviously an
abundance of observations randomly clumped at maximum light would bias the
mean to too bright a magnitude. Too many observations wasted at minimum
light would bias the mean too lo√w. The equivalent sigma of an intrinsic variable
with a 1.0 mag amplitude is 1.0 12 mag or approximately ±0.30 mag. To drive
6 B.F. Madore and W.L. Freedman
thiserroronthemeandownto0.03magwouldthenrequireontheorderof100
observations! This was unthinkable for a Hubble Space Telescope project intent
on observing more than a dozen galaxies.
Before describing our finally adopted sampling strategy, and reasoning for
it, we note that in a more general context the detection and characterization of
time-dependentsignalshasalongandwellstudiedhistory,especiallyinelectrical
engineeringanditsalliedsciences.Itwillnotberepeatedhere.However,wenote
that much of the theory and many of the practical applications involve equal-
spacedsamplingathighsignaltonoise,obtainedoverlongruntimes,andusually
covering many cycles of the searched-for, but unknown, signal. After detection,
the characterization or parameterization usually involves the determination of
a period, an amplitude and phase, and then finally the quantification of some
shape parameters of the signal. With regard to the latter point, we note that
periodic signals certainly are not always sinusoidal in form, but often they can
reasonably be decomposed into the superposition of a few low-order Fourier
components. Here we discuss an extreme corner of parameter space in signal
detection not much explored by others: a region defined by very small numbers
of observations (a mere handful), all of which are non-uniformly placed in time,
covering at most a few cycles of highly asymmetric, but still periodic, signals.
1.4 A Figure of Merit
As with any parameter extraction it is useful to have a quantitative measure
for the goodness of the solution. We begin by stating that the ideal distribution
of points over the phase-folded waveform is that where the observations fall
equallyspacedoverthelightcurve,wherenoneareredundant(i.e.,nocoincident
observations). With this in mind we have devised the following figure of merit:
For a given number of observations (N) we first calculate a variance in phase
space
(cid:1)N
∆2 = (φ −φ )2
N i+1 i
i=1
whichforthecaseoftheidealsampling(i.e.,thatofuniformandnon-redundant
placement around the light curve) reduces to
(cid:1)N
∆2 = (1/N)2 =1/N
N,uniform
i=1
(cid:2)N
(where φ = φ +1.0, and (φ −φ ) = 1.0). For the actual resultant
N+1 1 i+1 i
i=1
sampling (phase-folded to a given period) the equivalent (realized) statistic ∆2
N
is analogously derived from the sum of the squares of intervals separating the
orderpairsofobservationsovertheunitinterval.Thefinalfigureofmeritisthen
the difference between the realized ∆2 statistic and the ideal phasing statistic
N
1 The HST Key Project on the Extragalactic Distance Scale 7
∆2 , normalized by the ideal case, giving
N,uniform
U2 =[∆2 −∆2 ]/[(N −1)∆2 ]
N N,uniform N,uniform
or
(cid:1)N
U2 =[N (φ −φ )2−1]/(N −1)
i+1 i
i=1
Interpreted as a normalized variance, the U2 statistic has a value of zero when
thedistributionofpointsisnon-redundantlyuniformoverthelightcurve,anda
value of unity when all points are coincident in phase space. The added division
by(N-1)isintroducedtoforcethevariancetounityindependentofsamplesize,
whenallpointsclusteratasinglephase(e.g.,totalredundancy).Inthefollowing
however, we chose to plot the Uniformity Index (UI) which is based on U2 but
simply inverts and maps it onto the interval [0-100] by the following simple
transformation: UI =100[1−U2]. In this way a score of 100 indicates perfectly
uniform sampling over the light curve, and 0 indicates total failure, resulting
fromcompleteredundancyinthephaseplacementoftheobservationswherethe
data points, folded over the period of the variable, all end up at precisely the
same phase point.
1.5 Sampling Strategies
While it is intuitively obvious that optimal sampling of a signal with known
frequency should be undertaken in such a way that no two observations overlap
in phase as seen by the signal, what is perhaps not so obvious is that such
a uniform sampling strategy has very important consequences for that rate of
convergence of such things as the measured amplitude and the error on the
calculatedmean(bothofw√hichareintimatelyrelated).Unlikerandomsampling
which is a random-walk 1/ N process, uniform sampling has both of its errors
on amplitude and on the calculated mean drop much more rapidly; in fact,
those errors fall directly as 1/N. This is a considerable savings when additional
observations come at a high premium. A factor of 3 in observing time is always
easier to come by than is a factor of 10. Optimization of the type discussed here
buys that difference.
In the following series of plots and diagrams (Figs.1.3–1.9) we first explore
the systematics of the random sampling of highly asymmetric light curves using
extremely few observations. Indeed we begin with 2, 3 and 4 observations only
and then jump to 12 observations, which is the adopted number of observations
typically used in the small observing window imposed by orbital constraints
upon the Key project. The important details of the simulations are given in
extensive captions to the figures. The simulations bear out the expectation that
random sampling is highly inefficient in its error convergence properties where
the clearly Gaussian distribution of er√rors quantities of interest (such as the
mean magnitude) only go down like 1/ N, and where the first-order measure
of the light curve shape, the measured amplitude has a distribution that can be
8 B.F. Madore and W.L. Freedman
Fig.1.3. Monte Carlo Simulations of the mean magnitude, the uniformity index
andthederivedamplitudeforasyntheticlightcurveapproximatingthatofavariable
star.Thispanelshowsthedistributionsandmarginalizedvaluesforrandomsamplesof
twoobservations(N=2)only.Intheuppermiddlepanelisshownanexpandedviewof
the correlation of the mean magnitude versus the uniformity index. The right-handed
square bracket near UI = 1 shows the total range of mean magnitudes predicted for
the equivalent number of observations uniformly sampling the light curve (but with
random phase with respect to the periodic function). The next error bar to the right
uses the simulated data and shows the data-derived standard deviation (thick line)
and two-sigma (thin extension). The final error bar to the far right is the one-sigma
error bar for the uniform sampling. It can be shown that the distribution function for
themarginalizedmeans(centralpanel)forN=2isasymmetricaltriangularfunction
centered on a mean of 0.5. The marginalized distribution of observed amplitudes is a
ramp function with the modal value of the amplitude being zero
provenusingorderstatisticstohaveexactlytheformoftheBetafunctionwhich
only slowly converges on the true amplitude and has a long tail toward small
(derived) amplitudes.
Knowing that random sampling is too inefficient and that uniform sampling
over the light curve is ideal the problem becomes that of finding an observing
strategy (in real time) that corresponds to uniform sampling as viewed by a
1 The HST Key Project on the Extragalactic Distance Scale 9
Fig.1.4. Monte Carlo Simulations of the mean magnitude, the uniformity index
andthederivedamplitudeforasyntheticlightcurveapproximatingthatofavariable
star. This panel shows the distributions and marginalized values for random samples
of three observations (N = 3) only. The error bars are as discussed in Fig. 1.3; but
note again how much smaller the error bar for the uniform sampling is (far right)
as compared to the one-sigma error seen for random sampling (thick error bar). The
distribution of marginalized means (central panel) is rapidly becoming Gaussian in
appearance,whilethemarginalizeddistributionofamplitudesisverysymmetricabout
Amplitude = 0.5
Fig.1.5. The same as Figs. 1.3 and 1.4 except that this panel shows the results of
MonteCarlosimulationsofdistributionsandmarginalizedvaluesforrandomsampling
ofalightcurveusingonlyfourobservations(N=4).Themarginalizedmeanscontinue
tobecomemoreGaussianintheirdistribution,whilethedistributionofamplitudesand
uniformity indices are both markedly asymmetric, and skewed towards larger values
10 B.F. Madore and W.L. Freedman
Fig.1.6. Uniform Sampling in Time. The upper panel on the left shows the Uni-
formity Index (UI) as a function of period resulting from a sample of 12 observations
placed equidistantly within an observing window W = 100 days. As a function of pe-
riod(movingupinthepanel)onecanseethatthemeanleveloftheUniformityIndex
(asmarkedbythesolidverticallines)isadecreasingfunctionofperiod.Similarly,the
variance in the Uniformity Index increase toward shorter periods. Excursions to low
values of UI occur when data points fall redundantly at the same phase in the light
curve.
Themiddlepanelshowsaselectionoflightcurvesforperiodsrangingfromafewdays
upto80days.Theactualtimeoftheobservationwithinthe100-daywindowisshown
by the 12 vertical lines crossing the light curves at points marked by encircled dots.
TheUIforthe80-dayvariable(top)isveryhighandascanbeseenintherightpanel
thephase-foldedlightcurveisveryuniformlysampled.The36-dayvariableisalsouni-
formly sampled but its UI is significantly lower given the fact that each plotted point
represents three (overlapping) observations in the phase-folded plot. The 22-day Cep-
heid also shows redundant (phase-clumped) observations when folded over the known
period.The10-dayvariableisalsoveryuniformlycoveredbutastrongaliascanbecan
be readily seen in the real-time plot of the observations in the middle panel where an
equally good light curve having a period of about 100 days would produce an equally
compelling fit