Kochanek (1992) suggested that the redshifts of gravitational lens galaxies
rule out a large cosmological constant. This result was questioned by Helbig &
Kayser (1996), who pointed out that selection effects related to the brightness
of the lens can bias the results of this test against a high lambda value;
however, we did not claim that the observations favoured a high lambda value,
merely that current observational data were not sufficient to say either way,
using the test as proposed by Kochanek (1992) but corrected for selection
effects. Kochanek (1996) pointed out that additional information (fraction of
measured lens redshifts) provides additional information which restores the
sensitivity of the test to the cosmological model, at least somewhat. Here, I
consider three aspects. First, I examine the accuracy of the correction to the
test proposed by Kochanek (1996). Second, I compare the slightly different
statistical methods which have been used in connection with this test. Third, I
discuss what results can be obtained today now that more and better-defined
observations are available.Comment: 4 pages, LaTeX, 2 included PostScript files; to appear in the
  proceedings of the XXXVth Rencontres de Moriond, "L2K - Cosmological Physics
  with Gravitational Lensing", J.-P. Kneib, Y. Mellier, M. Moniez & J. Tran
  Thanh Van (eds.); see also http://terapix.iap.fr/L2K/l2k_one.htm

Helbig, Phillip

English

arXiv

Helbig, P

University of Groningen

   University of GroningenThe Lens-Redshift Test RevisitedHelbig, PPublished in:Default journalIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Helbig, P. (2000). The Lens-Redshift Test Revisited. Default journal.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019THE LENS-REDSHIFT TEST REVISITEDP. HELBIGRijksuniversiteit Groningen, Kapteyn Instituut, Postbus 800,NL-9700 AV Groningen, The NetherlandsKochanek1 suggested that the redshifts of gravitational lens galaxies rule out a large cosmolog-ical constant. This result was questioned by Helbig & Kayser2, who pointed out that selectioneects related to the brightness of the lens can bias the results of this test against a high λ0value; however, we did not claim that the observations favoured a high λ0 value, merely thatcurrent observational data were not sucient to say either way, using the test as proposedby Kochanek1 but corrected for selection eects. Kochanek3 pointed out that an additionalobservable, namely, the fraction of measured lens redshifts, provides additional informationwhich restores the sensitivity of the test to the cosmological model, at least somewhat. Here,I consider three aspects. First, I discuss the appropriateness of the correction to the testproposed by Kochanek (1996a). Second, I briefly mention the slightly dierent statisticalmethods which have been used in connection with this test. Third, I discuss what results canbe obtained today now that more and better-dened observations are available.1 IntroductionThe optical depth for gravitational lensing depends on the cosmological model, the Faber-Jackson and Tully-Fisher relations, lens-galaxy type (or the morphological mix), the luminosityfunction of lens galaxies and the S-z relation of the source population (e.g. Kochanek1, Helbig& Kayser2). There is an obvious problem with simply measuring the integrated optical depth,i.e. the number of lens systems (according to some useful denition): There is a degeneracybetween various parameters such that quite dierent combinations can result in the same num-ber of lenses. While it is possible to break this degeneracy somewhat, this requires a carefulsurvey and cannot be done with a sample of lenses ‘from the literature’. Kochanek1 pointed outthat one could use the shape of the optical-depth function dτ/dz as a probe of the cosmologicalmodel. The advantage of this approach is that it does not depend on the overall normalisation,as counting the number of lenses obviously does. Also, it is quite sensitive to the cosmologicalmodel, with the dependence on the cosmological model of a) the combination of angular sizedistances and b) the volume element, both of which appear in dτ/dz, reinforcing one another.In other words, the redshifts of lens galaxies can be used as a probe of the cosmological modelwhich is relatively little aected by our ignorance of other factors which determine the totaloptical depth.2 HistoryKochanek1 used a sample of 4 gravitational lens systems from the literature (estimating the lensredshift from absorption lines if unknown) and found that the Einstein-de Sitter model was 5{10times more likely than a flat model dominated by a cosmological constant. Helbig & Kayser2pointed out that this is potentially subject to a strong bias: It could be that most known lensredshifts are low not because we live in a universe in which this is more probable, but sincewe could not have measured them if they were higher. To correct for this eect, we suggestedcomparing the shape of dτ/dz not over the whole range [0,zs] (in practice, the value of thisfunction is negligible before zs is reached), but rather only out to that redshift where a lensredshift could have been measured, assuming some realistic limiting magnitude (at this redshift,dτ/dz usually still has a non-negligible value) and found that no interesting constraints couldbe obtained from then-current data (using 6 systems, all with measured, not estimated, lensredshifts), even if many more such systems were found, and that this conclusion did not dependon the precise value assumed for the limiting magnitude.Kochanek3 then pointed out that one can use an additional observable to restore cosmologicalsensitivity to the lens-redshift test: the fraction of lens systems with measured redshifts. If astrong bias were present such that only low lens redshifts could be measured, then there shouldbe many lens systems with unmeasured redshifts. While true, this misses the point of Helbig &Kayser2: Our claim was not that the observations supported a large value of the cosmologicalconstant (nor the opposite), but rather that the conclusion of Kochanek1 did not follow fromthe sample used (or our sample) since the lens-brightness bias had not been taken into account.Also, the correction proposed in Kochanek1 assumes that unknown lens redshifts are unknownonly because they are faint; in practice, there can be many other reasons why some lens redshiftshave not yet been measured (e.g. the maximum declination accessible from UKIRT).Various dierent statistical measures have been used to compare the observed and pre-dicted lens-redshift distributions. Here, I only consider the maximum-likelihood method (e.g.Kochanek3), which I consider to be most appropriate. However, results from using the binningmethod of Helbig & Kayser2 or a Kolmogorov-Smirnov test (Helbig, unpublished) give qualita-tively similar results.3 Using CLASSThe whole issue of unknown lens redshifts and their possible causes can be avoided if one hasa sample which is complete with respect to lens redshifts. CLASS (e.g. Helbig4) is close to thisgoal, and the JVAS subset of CLASS (more exactly, the JVAS lens systems in CLASS which arealso part of the statistically complete lens-survey sample; see Helbig4 for more details) is actuallycomplete. While only consisting of four systems, this is the same number used in Kochanek1, sothe time is ripe to revisit this topic. (The last JVAS lens redshift was obtained by Kochanek &Tonry5.)Figure 1 shows the likelihood as a function of λ0 and Ω0 for the sample from Kochanek1 whileFig. 2 shows the same for the JVAS lens systems B0218+357, MG0414+054, B1030+074 andB1422+231. It is obvious that the Kochanek1 sample indicates that the Einstein-de Sitter modelis more likely than a flat model dominated by a cosmological constant. The JVAS sample tellsa dierent story. Probably, part of the dierence, in particular, the low probability of modelsl  W-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.10.20.30.40.50.60.70.80.9Figure 1: Likelihood as a function of λ0 and Ω0 using the Kochanek sample; darker means higher likelihood.l  W-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.10.20.30.40.50.60.70.80.9Figure 2: Likelihood as a function of λ0 and Ω0 using the JVAS sample; darker means higher likelihood.near the white area to the lower right (which corresponds to no-big-bang models and is excludeda priori) can be explained by the bias noted in Helbig & Kayser2, while part can be explainedby small-number statistics. This will be explored in more detail in Helbig & Rusin6. (It shouldbe noted that the results for the Kochanek1 sample presented here do not correspond exactly tothose in Kochanek1 since there (as in Helbig & Kayser2), the now-known-to-be-erroneous (3/2)12factor for elliptical galaxies was used. Including this factor increases the relative likelihood ofthe Einstein-de Sitter model for the Kochanek1 sample while its eect on the JVAS sample isless pronounced.)4 Conclusions and Future ProspectsIt is obvious that the conclusion of Kochanek1 was premature: using a better-dened and in par-ticular bias-free (since complete) sample, the lens-redshift test does not disfavour cosmological-constant dominated models, although the signicance of this is not yet clear. Since the publica-tion of Kochanek1, of course, the cosmological constant has become popular again and, althoughmore detailed lens-statistics analyses are not incompatible with this (e.g. Helbig7), it is not yetclear whether systematic eects, such as our lack of sucient information about the S-z plane ofthe source population (e.g. Kochanek8), make current estimates of λ0 from the analysis of lenssurveys unreliable. It is at least interesting that the lens-redshift test does not seem to favouran Einstein-de Sitter universe over a model (flat or not) dominated by a cosmological constant.When the much larger CLASS sample is complete with respect to lens redshifts, the time willbe ripe to revisit this topic once again.AcknowledgmentsIt is a pleasure to thank David Rusin for useful discussions and the CLASS team for providingthe numbers to work with.References1. C.S. Kochanek, ApJ 384, 1 (1992)2. P. Helbig & R. Kayser, A&A 308, 359 (1996)3. C.S. Kochanek, ApJ 466, 638 (1996)4. P. Helbig, these proceedings5. J.L. Tonry & C.S. Kochanek, AJ 117, 2034 (1999)6. P. Helbig & D. Rusin, in preparation7. P. Helbig A&A 350, 1 (1999)8. C.S. Kochanek, ApJ 473, 595 (1996)

The Lens-Redshift Test Revisited

Abstract: Kochanek (1992) suggested that the redshifts of gravitational lens galaxies rule out a large cosmological constant. This result was questioned by Helbig & Kayser (1996), who pointed out that selection effects related to the brightness of the lens can bias the results of this test against a high lambda value; however, we did not claim that the observations favoured a high lambda value, merely that current observational data were not sufficient to say either way, using the test as proposed by Kochanek (1992) but corrected for selection effects. Kochanek (1996) pointed out that additional information (fraction of measured lens redshifts) provides additional information which restores the sensitivity of the test to the cosmological model, at least somewhat. Here, I consider three aspects. First, I examine the accuracy of the correction to the test proposed by Kochanek (1996). Second, I compare the slightly different statistical methods which have been used in connection with this test. Third, I discuss what results can be obtained today now that more and better-defined observations are available.

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University of Groningen Research Database

  
 University of Groningen
The Lens-Redshift Test Revisited
Helbig, P
Published in:
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IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
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Publication date:
2000
Link to publication in University of Groningen/UMCG research database
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Helbig, P. (2000). The Lens-Redshift Test Revisited. Default journal.
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Download date: 11-02-2018
THE LENS-REDSHIFT TEST REVISITED
P. HELBIG
Rijksuniversiteit Groningen, Kapteyn Instituut, Postbus 800,
NL-9700 AV Groningen, The Netherlands
Kochanek1 suggested that the redshifts of gravitational lens galaxies rule out a large cosmolog-
ical constant. This result was questioned by Helbig & Kayser2, who pointed out that selection
eects related to the brightness of the lens can bias the results of this test against a high λ0
value; however, we did not claim that the observations favoured a high λ0 value, merely that
current observational data were not sucient to say either way, using the test as proposed
by Kochanek1 but corrected for selection eects. Kochanek3 pointed out that an additional
observable, namely, the fraction of measured lens redshifts, provides additional information
which restores the sensitivity of the test to the cosmological model, at least somewhat. Here,
I consider three aspects. First, I discuss the appropriateness of the correction to the test
proposed by Kochanek (1996a). Second, I briefly mention the slightly dierent statistical
methods which have been used in connection with this test. Third, I discuss what results can
be obtained today now that more and better-dened observations are available.
1 Introduction
The optical depth for gravitational lensing depends on the cosmological model, the Faber-
Jackson and Tully-Fisher relations, lens-galaxy type (or the morphological mix), the luminosity
function of lens galaxies and the S-z relation of the source population (e.g. Kochanek1, Helbig
& Kayser2). There is an obvious problem with simply measuring the integrated optical depth,
i.e. the number of lens systems (according to some useful denition): There is a degeneracy
between various parameters such that quite dierent combinations can result in the same num-
ber of lenses. While it is possible to break this degeneracy somewhat, this requires a careful
survey and cannot be done with a sample of lenses ‘from the literature’. Kochanek1 pointed out
that one could use the shape of the optical-depth function dτ/dz as a probe of the cosmological
model. The advantage of this approach is that it does not depend on the overall normalisation,
as counting the number of lenses obviously does. Also, it is quite sensitive to the cosmological
model, with the dependence on the cosmological model of a) the combination of angular size
distances and b) the volume element, both of which appear in dτ/dz, reinforcing one another.
In other words, the redshifts of lens galaxies can be used as a probe of the cosmological model
which is relatively little aected by our ignorance of other factors which determine the total
optical depth.
2 History
Kochanek1 used a sample of 4 gravitational lens systems from the literature (estimating the lens
redshift from absorption lines if unknown) and found that the Einstein-de Sitter model was 5{10
times more likely than a flat model dominated by a cosmological constant. Helbig & Kayser2
pointed out that this is potentially subject to a strong bias: It could be that most known lens
redshifts are low not because we live in a universe in which this is more probable, but since
we could not have measured them if they were higher. To correct for this eect, we suggested
comparing the shape of dτ/dz not over the whole range [0,zs] (in practice, the value of this
function is negligible before zs is reached), but rather only out to that redshift where a lens
redshift could have been measured, assuming some realistic limiting magnitude (at this redshift,
dτ/dz usually still has a non-negligible value) and found that no interesting constraints could
be obtained from then-current data (using 6 systems, all with measured, not estimated, lens
redshifts), even if many more such systems were found, and that this conclusion did not depend
on the precise value assumed for the limiting magnitude.
Kochanek3 then pointed out that one can use an additional observable to restore cosmological
sensitivity to the lens-redshift test: the fraction of lens systems with measured redshifts. If a
strong bias were present such that only low lens redshifts could be measured, then there should
be many lens systems with unmeasured redshifts. While true, this misses the point of Helbig &
Kayser2: Our claim was not that the observations supported a large value of the cosmological
constant (nor the opposite), but rather that the conclusion of Kochanek1 did not follow from
the sample used (or our sample) since the lens-brightness bias had not been taken into account.
Also, the correction proposed in Kochanek1 assumes that unknown lens redshifts are unknown
only because they are faint; in practice, there can be many other reasons why some lens redshifts
have not yet been measured (e.g. the maximum declination accessible from UKIRT).
Various dierent statistical measures have been used to compare the observed and pre-
dicted lens-redshift distributions. Here, I only consider the maximum-likelihood method (e.g.
Kochanek3), which I consider to be most appropriate. However, results from using the binning
method of Helbig & Kayser2 or a Kolmogorov-Smirnov test (Helbig, unpublished) give qualita-
tively similar results.
3 Using CLASS
The whole issue of unknown lens redshifts and their possible causes can be avoided if one has
a sample which is complete with respect to lens redshifts. CLASS (e.g. Helbig4) is close to this
goal, and the JVAS subset of CLASS (more exactly, the JVAS lens systems in CLASS which are
also part of the statistically complete lens-survey sample; see Helbig4 for more details) is actually
complete. While only consisting of four systems, this is the same number used in Kochanek1, so
the time is ripe to revisit this topic. (The last JVAS lens redshift was obtained by Kochanek &
Tonry5.)
Figure 1 shows the likelihood as a function of λ0 and Ω0 for the sample from Kochanek1 while
Fig. 2 shows the same for the JVAS lens systems B0218+357, MG0414+054, B1030+074 and
B1422+231. It is obvious that the Kochanek1 sample indicates that the Einstein-de Sitter model
is more likely than a flat model dominated by a cosmological constant. The JVAS sample tells
a dierent story. Probably, part of the dierence, in particular, the low probability of models
l  
W
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
Figure 1: Likelihood as a function of λ0 and Ω0 using the Kochanek sample; darker means higher likelihood.
l
  
W
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
0.
8
0.
9
Figure 2: Likelihood as a function of λ0 and Ω0 using the JVAS sample; darker means higher likelihood.
near the white area to the lower right (which corresponds to no-big-bang models and is excluded
a priori) can be explained by the bias noted in Helbig & Kayser2, while part can be explained
by small-number statistics. This will be explored in more detail in Helbig & Rusin6. (It should
be noted that the results for the Kochanek1 sample presented here do not correspond exactly to
those in Kochanek1 since there (as in Helbig & Kayser2), the now-known-to-be-erroneous (3/2)
1
2
factor for elliptical galaxies was used. Including this factor increases the relative likelihood of
the Einstein-de Sitter model for the Kochanek1 sample while its eect on the JVAS sample is
less pronounced.)
4 Conclusions and Future Prospects
It is obvious that the conclusion of Kochanek1 was premature: using a better-dened and in par-
ticular bias-free (since complete) sample, the lens-redshift test does not disfavour cosmological-
constant dominated models, although the signicance of this is not yet clear. Since the publica-
tion of Kochanek1, of course, the cosmological constant has become popular again and, although
more detailed lens-statistics analyses are not incompatible with this (e.g. Helbig7), it is not yet
clear whether systematic eects, such as our lack of sucient information about the S-z plane of
the source population (e.g. Kochanek8), make current estimates of λ0 from the analysis of lens
surveys unreliable. It is at least interesting that the lens-redshift test does not seem to favour
an Einstein-de Sitter universe over a model (flat or not) dominated by a cosmological constant.
When the much larger CLASS sample is complete with respect to lens redshifts, the time will
be ripe to revisit this topic once again.
Acknowledgments
It is a pleasure to thank David Rusin for useful discussions and the CLASS team for providing
the numbers to work with.
References
1. C.S. Kochanek, ApJ 384, 1 (1992)
2. P. Helbig & R. Kayser, A&A 308, 359 (1996)
3. C.S. Kochanek, ApJ 466, 638 (1996)
4. P. Helbig, these proceedings
5. J.L. Tonry & C.S. Kochanek, AJ 117, 2034 (1999)
6. P. Helbig & D. Rusin, in preparation
7. P. Helbig A&A 350, 1 (1999)
8. C.S. Kochanek, ApJ 473, 595 (1996)


Abstract: Kochanek (1992) suggested that the redshifts of gravitational lens galaxies rule out a large cosmological constant. This result was questioned by Helbig &amp; Kayser (1996), who pointed out that selection effects related to the brightness of the lens can bias the results of this test against a high lambda value; however, we did not claim that the observations favoured a high lambda value, merely that current observational data were not sufficient to say either way, using the test as proposed by Kochanek (1992) but corrected for selection effects. Kochanek (1996) pointed out that additional information (fraction of measured lens redshifts) provides additional information which restores the sensitivity of the test to the cosmological model, at least somewhat. Here, I consider three aspects. First, I examine the accuracy of the correction to the test proposed by Kochanek (1996). Second, I compare the slightly different statistical methods which have been used in connection with this test. Third, I discuss what results can be obtained today now that more and better-defined observations are available

ARTS repository - University of Groningen

   University of GroningenThe Lens-Redshift Test RevisitedHelbig, PPublished in:ArXivIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionEarly version, also known as pre-printPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Helbig, P. (2000). The Lens-Redshift Test Revisited. ArXiv. https://arxiv.org/abs/astro-ph/0008198CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-10-2023THE LENS-REDSHIFT TEST REVISITEDP. HELBIGRijksuniversiteit Groningen, Kapteyn Instituut, Postbus 800,NL-9700 AV Groningen, The NetherlandsKochanek1 suggested that the redshifts of gravitational lens galaxies rule out a large cosmolog-ical constant. This result was questioned by Helbig & Kayser2, who pointed out that selectioneffects related to the brightness of the lens can bias the results of this test against a high λ0value; however, we did not claim that the observations favoured a high λ0 value, merely thatcurrent observational data were not sufficient to say either way, using the test as proposedby Kochanek1 but corrected for selection effects. Kochanek3 pointed out that an additionalobservable, namely, the fraction of measured lens redshifts, provides additional informationwhich restores the sensitivity of the test to the cosmological model, at least somewhat. Here,I consider three aspects. First, I discuss the appropriateness of the correction to the testproposed by Kochanek (1996a). Second, I briefly mention the slightly different statisticalmethods which have been used in connection with this test. Third, I discuss what results canbe obtained today now that more and better-defined observations are available.1 IntroductionThe optical depth for gravitational lensing depends on the cosmological model, the Faber-Jackson and Tully-Fisher relations, lens-galaxy type (or the morphological mix), the luminosityfunction of lens galaxies and the S-z relation of the source population (e.g. Kochanek1, Helbig& Kayser2). There is an obvious problem with simply measuring the integrated optical depth,i.e. the number of lens systems (according to some useful definition): There is a degeneracybetween various parameters such that quite different combinations can result in the same num-ber of lenses. While it is possible to break this degeneracy somewhat, this requires a carefulsurvey and cannot be done with a sample of lenses ‘from the literature’. Kochanek1 pointed outthat one could use the shape of the optical-depth function dτ/dz as a probe of the cosmologicalmodel. The advantage of this approach is that it does not depend on the overall normalisation,as counting the number of lenses obviously does. Also, it is quite sensitive to the cosmologicalmodel, with the dependence on the cosmological model of a) the combination of angular sizedistances and b) the volume element, both of which appear in dτ/dz, reinforcing one another.In other words, the redshifts of lens galaxies can be used as a probe of the cosmological modelwhich is relatively little affected by our ignorance of other factors which determine the totaloptical depth.2 HistoryKochanek1 used a sample of 4 gravitational lens systems from the literature (estimating the lensredshift from absorption lines if unknown) and found that the Einstein-de Sitter model was 5–10times more likely than a flat model dominated by a cosmological constant. Helbig & Kayser2pointed out that this is potentially subject to a strong bias: It could be that most known lensredshifts are low not because we live in a universe in which this is more probable, but sincewe could not have measured them if they were higher. To correct for this effect, we suggestedcomparing the shape of dτ/dz not over the whole range [0,zs] (in practice, the value of thisfunction is negligible before zs is reached), but rather only out to that redshift where a lensredshift could have been measured, assuming some realistic limiting magnitude (at this redshift,dτ/dz usually still has a non-negligible value) and found that no interesting constraints couldbe obtained from then-current data (using 6 systems, all with measured, not estimated, lensredshifts), even if many more such systems were found, and that this conclusion did not dependon the precise value assumed for the limiting magnitude.Kochanek3 then pointed out that one can use an additional observable to restore cosmologicalsensitivity to the lens-redshift test: the fraction of lens systems with measured redshifts. If astrong bias were present such that only low lens redshifts could be measured, then there shouldbe many lens systems with unmeasured redshifts. While true, this misses the point of Helbig &Kayser2: Our claim was not that the observations supported a large value of the cosmologicalconstant (nor the opposite), but rather that the conclusion of Kochanek1 did not follow fromthe sample used (or our sample) since the lens-brightness bias had not been taken into account.Also, the correction proposed in Kochanek1 assumes that unknown lens redshifts are unknownonly because they are faint; in practice, there can be many other reasons why some lens redshiftshave not yet been measured (e.g. the maximum declination accessible from UKIRT).Various different statistical measures have been used to compare the observed and pre-dicted lens-redshift distributions. Here, I only consider the maximum-likelihood method (e.g.Kochanek3), which I consider to be most appropriate. However, results from using the binningmethod of Helbig & Kayser2 or a Kolmogorov-Smirnov test (Helbig, unpublished) give qualita-tively similar results.3 Using CLASSThe whole issue of unknown lens redshifts and their possible causes can be avoided if one hasa sample which is complete with respect to lens redshifts. CLASS (e.g. Helbig4) is close to thisgoal, and the JVAS subset of CLASS (more exactly, the JVAS lens systems in CLASS which arealso part of the statistically complete lens-survey sample; see Helbig4 for more details) is actuallycomplete. While only consisting of four systems, this is the same number used in Kochanek1, sothe time is ripe to revisit this topic. (The last JVAS lens redshift was obtained by Kochanek &Tonry5.)Figure 1 shows the likelihood as a function of λ0 and Ω0 for the sample from Kochanek1 whileFig. 2 shows the same for the JVAS lens systems B0218+357, MG0414+054, B1030+074 andB1422+231. It is obvious that the Kochanek1 sample indicates that the Einstein-de Sitter modelis more likely than a flat model dominated by a cosmological constant. The JVAS sample tellsa different story. Probably, part of the difference, in particular, the low probability of modelsl  W-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.10.20.30.40.50.60.70.80.9Figure 1: Likelihood as a function of λ0 and Ω0 using the Kochanek sample; darker means higher likelihood.l  W-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.10.20.30.40.50.60.70.80.9Figure 2: Likelihood as a function of λ0 and Ω0 using the JVAS sample; darker means higher likelihood.near the white area to the lower right (which corresponds to no-big-bang models and is excludeda priori) can be explained by the bias noted in Helbig & Kayser2, while part can be explainedby small-number statistics. This will be explored in more detail in Helbig & Rusin6. (It shouldbe noted that the results for the Kochanek1 sample presented here do not correspond exactly tothose in Kochanek1 since there (as in Helbig & Kayser2), the now-known-to-be-erroneous (3/2)12factor for elliptical galaxies was used. Including this factor increases the relative likelihood ofthe Einstein-de Sitter model for the Kochanek1 sample while its effect on the JVAS sample isless pronounced.)4 Conclusions and Future ProspectsIt is obvious that the conclusion of Kochanek1 was premature: using a better-defined and in par-ticular bias-free (since complete) sample, the lens-redshift test does not disfavour cosmological-constant dominated models, although the significance of this is not yet clear. Since the publica-tion of Kochanek1, of course, the cosmological constant has become popular again and, althoughmore detailed lens-statistics analyses are not incompatible with this (e.g. Helbig7), it is not yetclear whether systematic effects, such as our lack of sufficient information about the S-z plane ofthe source population (e.g. Kochanek8), make current estimates of λ0 from the analysis of lenssurveys unreliable. It is at least interesting that the lens-redshift test does not seem to favouran Einstein-de Sitter universe over a model (flat or not) dominated by a cosmological constant.When the much larger CLASS sample is complete with respect to lens redshifts, the time willbe ripe to revisit this topic once again.AcknowledgmentsIt is a pleasure to thank David Rusin for useful discussions and the CLASS team for providingthe numbers to work with.References1. C.S. Kochanek, ApJ 384, 1 (1992)2. P. Helbig & R. Kayser, A&A 308, 359 (1996)3. C.S. Kochanek, ApJ 466, 638 (1996)4. P. Helbig, these proceedings5. J.L. Tonry & C.S. Kochanek, AJ 117, 2034 (1999)6. P. Helbig & D. Rusin, in preparation7. P. Helbig A&A 350, 1 (1999)8. C.S. Kochanek, ApJ 473, 595 (1996)

   University of GroningenThe Lens-Redshift Test RevisitedHelbig, PPublished in:ArXivIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionEarly version, also known as pre-printPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Helbig, P. (2000). The Lens-Redshift Test Revisited. ArXiv. https://arxiv.org/abs/astro-ph/0008198CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 08-06-2022THE LENS-REDSHIFT TEST REVISITEDP. HELBIGRijksuniversiteit Groningen, Kapteyn Instituut, Postbus 800,NL-9700 AV Groningen, The NetherlandsKochanek1 suggested that the redshifts of gravitational lens galaxies rule out a large cosmolog-ical constant. This result was questioned by Helbig & Kayser2, who pointed out that selectioneffects related to the brightness of the lens can bias the results of this test against a high λ0value; however, we did not claim that the observations favoured a high λ0 value, merely thatcurrent observational data were not sufficient to say either way, using the test as proposedby Kochanek1 but corrected for selection effects. Kochanek3 pointed out that an additionalobservable, namely, the fraction of measured lens redshifts, provides additional informationwhich restores the sensitivity of the test to the cosmological model, at least somewhat. Here,I consider three aspects. First, I discuss the appropriateness of the correction to the testproposed by Kochanek (1996a). Second, I briefly mention the slightly different statisticalmethods which have been used in connection with this test. Third, I discuss what results canbe obtained today now that more and better-defined observations are available.1 IntroductionThe optical depth for gravitational lensing depends on the cosmological model, the Faber-Jackson and Tully-Fisher relations, lens-galaxy type (or the morphological mix), the luminosityfunction of lens galaxies and the S-z relation of the source population (e.g. Kochanek1, Helbig& Kayser2). There is an obvious problem with simply measuring the integrated optical depth,i.e. the number of lens systems (according to some useful definition): There is a degeneracybetween various parameters such that quite different combinations can result in the same num-ber of lenses. While it is possible to break this degeneracy somewhat, this requires a carefulsurvey and cannot be done with a sample of lenses ‘from the literature’. Kochanek1 pointed outthat one could use the shape of the optical-depth function dτ/dz as a probe of the cosmologicalmodel. The advantage of this approach is that it does not depend on the overall normalisation,as counting the number of lenses obviously does. Also, it is quite sensitive to the cosmologicalmodel, with the dependence on the cosmological model of a) the combination of angular sizedistances and b) the volume element, both of which appear in dτ/dz, reinforcing one another.In other words, the redshifts of lens galaxies can be used as a probe of the cosmological modelwhich is relatively little affected by our ignorance of other factors which determine the totaloptical depth.2 HistoryKochanek1 used a sample of 4 gravitational lens systems from the literature (estimating the lensredshift from absorption lines if unknown) and found that the Einstein-de Sitter model was 5–10times more likely than a flat model dominated by a cosmological constant. Helbig & Kayser2pointed out that this is potentially subject to a strong bias: It could be that most known lensredshifts are low not because we live in a universe in which this is more probable, but sincewe could not have measured them if they were higher. To correct for this effect, we suggestedcomparing the shape of dτ/dz not over the whole range [0,zs] (in practice, the value of thisfunction is negligible before zs is reached), but rather only out to that redshift where a lensredshift could have been measured, assuming some realistic limiting magnitude (at this redshift,dτ/dz usually still has a non-negligible value) and found that no interesting constraints couldbe obtained from then-current data (using 6 systems, all with measured, not estimated, lensredshifts), even if many more such systems were found, and that this conclusion did not dependon the precise value assumed for the limiting magnitude.Kochanek3 then pointed out that one can use an additional observable to restore cosmologicalsensitivity to the lens-redshift test: the fraction of lens systems with measured redshifts. If astrong bias were present such that only low lens redshifts could be measured, then there shouldbe many lens systems with unmeasured redshifts. While true, this misses the point of Helbig &Kayser2: Our claim was not that the observations supported a large value of the cosmologicalconstant (nor the opposite), but rather that the conclusion of Kochanek1 did not follow fromthe sample used (or our sample) since the lens-brightness bias had not been taken into account.Also, the correction proposed in Kochanek1 assumes that unknown lens redshifts are unknownonly because they are faint; in practice, there can be many other reasons why some lens redshiftshave not yet been measured (e.g. the maximum declination accessible from UKIRT).Various different statistical measures have been used to compare the observed and pre-dicted lens-redshift distributions. Here, I only consider the maximum-likelihood method (e.g.Kochanek3), which I consider to be most appropriate. However, results from using the binningmethod of Helbig & Kayser2 or a Kolmogorov-Smirnov test (Helbig, unpublished) give qualita-tively similar results.3 Using CLASSThe whole issue of unknown lens redshifts and their possible causes can be avoided if one hasa sample which is complete with respect to lens redshifts. CLASS (e.g. Helbig4) is close to thisgoal, and the JVAS subset of CLASS (more exactly, the JVAS lens systems in CLASS which arealso part of the statistically complete lens-survey sample; see Helbig4 for more details) is actuallycomplete. While only consisting of four systems, this is the same number used in Kochanek1, sothe time is ripe to revisit this topic. (The last JVAS lens redshift was obtained by Kochanek &Tonry5.)Figure 1 shows the likelihood as a function of λ0 and Ω0 for the sample from Kochanek1 whileFig. 2 shows the same for the JVAS lens systems B0218+357, MG0414+054, B1030+074 andB1422+231. It is obvious that the Kochanek1 sample indicates that the Einstein-de Sitter modelis more likely than a flat model dominated by a cosmological constant. The JVAS sample tellsa different story. Probably, part of the difference, in particular, the low probability of modelsl  W-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.10.20.30.40.50.60.70.80.9Figure 1: Likelihood as a function of λ0 and Ω0 using the Kochanek sample; darker means higher likelihood.l  W-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.10.20.30.40.50.60.70.80.9Figure 2: Likelihood as a function of λ0 and Ω0 using the JVAS sample; darker means higher likelihood.near the white area to the lower right (which corresponds to no-big-bang models and is excludeda priori) can be explained by the bias noted in Helbig & Kayser2, while part can be explainedby small-number statistics. This will be explored in more detail in Helbig & Rusin6. (It shouldbe noted that the results for the Kochanek1 sample presented here do not correspond exactly tothose in Kochanek1 since there (as in Helbig & Kayser2), the now-known-to-be-erroneous (3/2)12factor for elliptical galaxies was used. Including this factor increases the relative likelihood ofthe Einstein-de Sitter model for the Kochanek1 sample while its effect on the JVAS sample isless pronounced.)4 Conclusions and Future ProspectsIt is obvious that the conclusion of Kochanek1 was premature: using a better-defined and in par-ticular bias-free (since complete) sample, the lens-redshift test does not disfavour cosmological-constant dominated models, although the significance of this is not yet clear. Since the publica-tion of Kochanek1, of course, the cosmological constant has become popular again and, althoughmore detailed lens-statistics analyses are not incompatible with this (e.g. Helbig7), it is not yetclear whether systematic effects, such as our lack of sufficient information about the S-z plane ofthe source population (e.g. Kochanek8), make current estimates of λ0 from the analysis of lenssurveys unreliable. It is at least interesting that the lens-redshift test does not seem to favouran Einstein-de Sitter universe over a model (flat or not) dominated by a cosmological constant.When the much larger CLASS sample is complete with respect to lens redshifts, the time willbe ripe to revisit this topic once again.AcknowledgmentsIt is a pleasure to thank David Rusin for useful discussions and the CLASS team for providingthe numbers to work with.References1. C.S. Kochanek, ApJ 384, 1 (1992)2. P. Helbig & R. Kayser, A&A 308, 359 (1996)3. C.S. Kochanek, ApJ 466, 638 (1996)4. P. Helbig, these proceedings5. J.L. Tonry & C.S. Kochanek, AJ 117, 2034 (1999)6. P. Helbig & D. Rusin, in preparation7. P. Helbig A&A 350, 1 (1999)8. C.S. Kochanek, ApJ 473, 595 (1996)

Kochanek (1992) suggested that the redshifts of gravitational lens galaxies rule out a large cosmological constant. This result was questioned by Helbig & Kayser (1996), who pointed out that selection effects related to the brightness of the lens can bias the results of this test against a high lambda value; however, we did not claim that the observations favoured a high lambda value, merely that current observational data were not sufficient to say either way, using the test as proposed by Kochanek (1992) but corrected for selection effects. Kochanek (1996) pointed out that additional information (fraction of measured lens redshifts) provides additional information which restores the sensitivity of the test to the cosmological model, at least somewhat. Here, I consider three aspects. First, I examine the accuracy of the correction to the test proposed by Kochanek (1996). Second, I compare the slightly different statistical methods which have been used in connection with this test. Third, I discuss what results can be obtained today now that more and better-defined observations are available

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Abstract: Kochanek (1992) suggested that the redshifts of gravitational lens galaxies rule out a large cosmological constant. This result was questioned by Helbig & Kayser (1996), who pointed out that selection effects related to the brightness of the lens can bias the results of this test against a high lambda value; however, we did not claim that the observations favoured a high lambda value, merely that current observational data were not sufficient to say either way, using the test as proposed by Kochanek (1992) but corrected for selection effects. Kochanek (1996) pointed out that additional information (fraction of measured lens redshifts) provides additional information which restores the sensitivity of the test to the cosmological model, at least somewhat. Here, I consider three aspects. First, I examine the accuracy of the correction to the test proposed by Kochanek (1996). Second, I compare the slightly different statistical methods which have been used in connection with this test. Third, I discuss what results can be obtained today now that more and better-defined observations are available

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The Lens-Redshift Test Revisited

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