1. The Alcock-Paczynski Probe of Cosmology Lyman-  forest, LSS, And Cosmic Consistency Dark Energy Workshop Center for Cosmological Physics University of Chicago 15 December 2001 Albert Stebbins Fermilab

2. STEBBINS: Alcock-Paczynski 2 The Alcock-Paczynski Test Alcock & Paczynski (1979) An evolution free test for non-zero cosmological constant Nature 281 358 There are 2 directly observable measures of the size of an “object” expanding w/ cosmological flow Angular size Radial extent in redshift space If such objects are not preferentially aligned either along or perpendicular to our line-of-sight then, by requiring no such preferential alignment, one can determine the ratio of the conversion factors, angular distance to physical distance, to that from redshift distance to physical distance. i.e. one can determine  z/  [z].  z

3. STEBBINS: Alcock-Paczynski 3 Observational Fundamentalism Many cosmological tests make use of only two functions of redshift the angular diameter distance (comoving or physical) the radial comoving distance Furthermore these two functions are not independent, as they are related by the relation (which we refer to as cosmic consistency) Where S s [x]=x, sin[x], sinh[x] for s=0,+1,-1, respectively; and K is the spatial curvature Thus there are only so many independent cosmological tests! This is good as one measurement checks another! We are measuring different things at present only because we are measuring different redshift regimes e.g, CMB z~1000, SNeIa z~0.5, lensing z~1.

4. STEBBINS: Alcock-Paczynski 4 Different Tests - Different Combinations Examples of how these two functions are related to standard tests the apparent luminosity of standard candles (the “K-correction”, k[z], includes (1+z) 4 surface brightness dimming and redshift of spectrum into /out of observational band) the cosmological volume element (  # of objects) per unit redshift per unit solid angle the Alcock-Paczynski test N.B. in practice other cosmological dependencies tend to creep into these tests, e.g. the linear growth rate of perturbations, or more complicated things like the star formation rate.

5. STEBBINS: Alcock-Paczynski 5 A-P: Just Another Cosmological Test As with all such tests one must go to significant redshift to measure anything interesting. For z<<0 you already know what the answer is. The AP test at lo-z quickly (z>0.5) becomes sensitive to the presence of , but only at the 20% level. It is insensitive to curvature at lo-z, rather like the SNeIa l[z]. At very hi-z it becomes most sensitive to the curvature at about the 70% level. At hi-z it is relatively insensitive to , rather like the CMB l peak test.

6. STEBBINS: Alcock-Paczynski 6 A-P: Just The Same Cosmological Test

7. STEBBINS: Alcock-Paczynski 7 Cosmological Consistency As described, the results of different cosmological tests are inter-related. Some of these relationships are “axiomatic”, e.g. Other relationship depend on the cosmic consistency relation, e.g. Which relates observables from an A-P test and a l-z (e.g. SNeIa) test to Which probably isn’t quite measurable. However since the right-hand-side is z-independent one can test cosmic consistency by requiring that one infers the same  0 at each z.

8. STEBBINS: Alcock-Paczynski 8 ¿Cosmological Inconsistency? These relations hold no matter how weird the dark energy is! Violation of an axiomatic relation probably indicates a measurement error or mis- interpretation of measurements. The cosmic consistency relations is a result of assumptions of the FRW (Friedmann-Robertson-Walker Cosmology - one of the fundamental tenets upon which interpretation of cosmological observations is based. Violation of cosmic consistency might indicate 1.non-FRW geometry i.e. we live in the center of a spherically symmetric but non- homogeneous universe (violation of cosmological Copernican Principle) 2.non-metric theory for propagation of light (post-modern tired light) - as we are in a sense measuring the metric with these tests. 3.Measurement error or a problem with interpretation of measurements. As the relations combine different tests, and as it is unlikely that errors in one test would balance errors in another such as to satisfy the relations, this provides a powerful check of all tests involved! It is thus worthwhile to compare the AP test at the same redshifts as SNeIa

9. STEBBINS: Alcock-Paczynski 9 Alcock-Paczynski Realities Systematics Since angular size is measure of physical size and radial size measure of velocity differences we do expect that the two are the same - there can be preferential alignment w.r.t. line-of- sight: i.e. redshift space distortions. These distortions must be understood and taken into account. On small scales ≤1 Mpc astrophysical objects have separated from cosmic expansion and have little to do w/ cosmic expansion (fingers of God). On large scales >20 Mpc (@z=0) simple linear theory distortions (Kaiser effect) may suffice. As with all cosmological tests one must overcome observational hurdles in order to make the test a useful one. Statistics If objects were truly round in redshift space then one need observe only one at each z to determine  z/  [z]. More generally accuracy is given by*  ln(  z/  )~e/√(8N) where N is the number of independent objects and e their RMS ellipticity. N.B. 0 ≤ e ≤ 1 Statistical measurement errors decreases effective N. * this result cribbed from weak lensing theory

10. STEBBINS: Alcock-Paczynski 10 Large Scale Structure: Voids, Filaments, etc. From galaxy redshift surveys one may identify structures such as voids (Ryden) or filaments (Möller & Fynbo), measure their shapes and use these for the AP test As these are quasi-linear structures the redshift space distortions are non- trivial to correct for. At present surveys dense enough to identify structures are at lo-z where the AP is less useful. In the future DEEP and VIRMOS will provide dense surveys at z~1. SDSS Galaxy Redshift Survey: Early Data Stoughton et al. (2001) Sloan Digital Sky Survey: Early Data Release in preparation

11. STEBBINS: Alcock-Paczynski 11 Large Scale Structure: Sparse Sampling Sparse surveys efficiently measure the 2-pt statistics of clustering especially on large scales where the perturbations are linear. They are not useful for identifying individual structures. e.g. the BRG (Bright Red Galaxy) part of the SDSS redshift survey, or much of 2DF. SDSS Galaxy Redshift Survey: Early Data Stoughton et al. (2001) Sloan Digital Sky Survey: Early Data Release in preparation

12. STEBBINS: Alcock-Paczynski 12 Large Scale Structure: QSOs Or the quasar redshift survey that is part of the SDSS (Calvão, De Mello Neto, Waga). SDSS Galaxy Redshift Survey: Early Data Stoughton et al. (2001) Sloan Digital Sky Survey: Early Data Release in preparation Schneider et al. (2001) The Sloan Digital Sky Survey Quasar Catalog I: Early Data Release astro-ph/0110629

13. STEBBINS: Alcock-Paczynski 13 Composite Objects:  [r p,  ] One may also use statistics of redshift space clustering in place of shapes of individual objects. In particular the redshift space 2- point function  [r p,  ]=  [ ,  z] This is a convenient way of combining all of the data w/o identifying objects. One can use this in cases where, say, the galaxy sampling is too sparse to allow accurate identification of objects. SDSS Galaxy Redshift-Space Correlation Zehavi et al. (2001) Galaxy Clustering in Early SDSS Redshift Data astro-ph/0106476

14. STEBBINS: Alcock-Paczynski 14 Alcock-Paczynski + Redshift Space Distortions Redshift space distortions themselves give some clue as to the cosmological parameters c.f. the Kaiser effect Combining the AP test w/ theory for redshift space distortion (and to some extent bias) one can obtain a combined constraint on cosmological parameters (Matsubara & Szalay). e.g. for SDSS Northern survey (Subbarao) OTHERS FIXED mm  b/mb/m hn 88 b Main ±3%±19%±16%±4%±2%±0.5% BRG ±2%±4%±9%±2%±1%±0.3%±0.4% QSO ±14%±15%±76%±20%±14%±5%±6% MARGIN- ALIZED mm  b/mb/m b Main ±14%±57%±51%±2% BRG ±9%±10%±33%±0.9% QSO ±170%±75%±360%±69% SDSS Parameter Estimation Matsubara & Szalay (2001) Constraining the Cosmological Constant from Large-Scale Redshift- Space Clustering astro-ph/0105493

15. STEBBINS: Alcock-Paczynski 15 Alcock-Paczynski + Redshift Space Distortions Parameter Estimation for 1. (200h -1 Mpc) 3 cube 2.  =  survey Matsubara & Szalay (2001) Constraining the Cosmological Constant from Large-Scale Redshift-Space Clustering astro-ph/0105493 Shot Noise: (20h -1 Mpc) 3 n=0.1, 0.3, 1, 3, 10, ∞

16. STEBBINS: Alcock-Paczynski 16 Lyman-  Forest Structure along line-of-sight to QSOs  z Continuum Fitting Systematic!

17. STEBBINS: Alcock-Paczynski 17 The Ly-  Alcock-Paczynski Forest Test McDonald & Miralda-Escudé (1999) Measuring the Cosmological Geometry from the Ly  Forest Along Parallel Lines of Sight Ap.J. 518 24 Hui, Stebbins, & Burles (1999) A Geometrical Test of the Cosmological Energy Contents Using the Lyman-alpha Forest Ap.J Lett. 511 L5  z

18. STEBBINS: Alcock-Paczynski 18 The Alcock-Paczynski Ly-  Forest Test  z The quality of the AP test depends on the QSO separation Too small Just right Too large

19. STEBBINS: Alcock-Paczynski 19 Alcock-Paczynski Test Applied to QSO Triplet Burles, Stebbins, & Hui (circa 1999, unpublished) In practice one cross- correlates the Ly-  absorption e -  between the different lines-of- sight. This can be done in space or it’s Fourier transform. One expects the correlation to be near perfect on z-scales larger than the transverse separation, no correlation on scales much larger than the separation, with roughly an exponential falloff. First Try

20. STEBBINS: Alcock-Paczynski 20 Ly-  Forest Sensitivity Cosmological Accuracy from SDSS QSOs McDonald (2001) Toward a Measurement of the Cosmological Geometry at z~2: Predicting Ly  Forest Correlations in Three Dimensions, and the Potential of Future Data Sets astro- ph/0108064 McDonald (2001) has performed detailed modeling of expected SDSS QSOs, comparing w/ simulations to model redshift space distortions. With followup spectra (i.e. apart from SDSS spectroscopy) one can obtain a respectable limit on .

21. STEBBINS: Alcock-Paczynski 21 Conclusions Alcock-Paczynski test - yet another cosmological test. Redundant with other tests in z ranges where they overlap. Implementations have been proposed up to QSO redshifts (z~3) - perhaps further w/ IR spectroscopy. No useful applications have yet been carried out. For deep redshift surveys - and when combined with theory of redshift space distortion - can provide very tight constraints on a[t] a.k.a. p[  ]. Probably provides best probe of cosmology at z~2 through Ly-  Forest.