the alcock-paczynski probe of cosmology lyman- forest, lss, and cosmic consistency dark energy...

The Alcock-Paczynski Probeof Cosmology

Lyman- forest, LSS,

And Cosmic Consistency

Dark Energy WorkshopCenter for Cosmological PhysicsUniversity of Chicago15 December 2001

Albert StebbinsFermilab

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

δzδθ

=13

δzδθ

=1

δzδθ

=3


Observational Fundamentalism

DA,co[z]≡(1+z)DA[z]

DR,co[z]

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 Ss[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.

DA,co[z]=1

|K | Ssgn[K ][K DR,co[z]]

K =(Ω0 −1)c2

H02


Different Tests - Different CombinationsExamples 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.

l[z]=k[z]L

DA[z]2

dVco

dz dΩ[z]=DA,co[z]

2DR,co′[z].

δzδθ

[z]=DA,co[z]

DR,co′[z]


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 lpeak test.


A-P: Just The Same Cosmological Test

0 1 2 3WM

- 1

0

1

2

3

WL

dzÄÄÄÄÄÄÄÄdq

=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7û z=0.5

NoBigBang

0 1 2 3WM

- 1

0

1

2

3

WL

10- 4 dzÄÄÄÄÄÄÄÄdq

=0.5, 1, 2, 3, 5, 10, 20, 30û z=1000

NoBigBang


Cosmological ConsistencyAs 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.

14

(δzδθ

[z])2(∂z ln[k[z] Ll[z]

])2 −1

(cH0

)2 k[z] Ll[z]

=1−Ω0

(δzδθ

[z])(dVdz dΩ

) =(k[z] Ll[z]

)32

Ω0 =ρb +ργ +ρDM +ρQ

ρcrit

ρcrit ≡3

8π GH0

2


¿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 indicate1. 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


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


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 DataStoughton et al. (2001) Sloan Digital Sky Survey: Early Data Releasein preparation


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 DataStoughton et al. (2001) Sloan Digital Sky Survey: Early Data Releasein preparation


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 DataStoughton et al. (2001) Sloan Digital Sky Survey: Early Data Releasein preparationSchneider et al. (2001) The Sloan Digital Sky Survey Quasar Catalog I: Early Data Release astro-ph/0110629


Composite Objects: [rp,]

• One may also use statistics of redshift space clustering in place of shapes of individual objects.

• In particular the redshift space 2-point function [rp,]= [ ,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


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)

PRS[k⊥,k||]=(1+βk||

k||2 +k⊥

2)2P3D[ k||

2+k⊥

2] β≅

Ωm0.6

b

OTHERS FIXED m b/m h n 8 b

Main ±3% ±19% ±16% ±4% ±2% ±0.5% ±0.5%

BRG ±2% ±4% ±9% ±2% ±1% ±0.3% ±0.4%

QSO ±14% ±15% ±76% ±20% ±14% ±5% ±6%

MARGIN- ALIZED m b/m b

Main ±14% ±57% ±51% ±2%

BRG ±9% ±10% ±33% ±0.9%

QSO ±170% ±75% ±360% ±69%

SDSS Parameter EstimationMatsubara & Szalay (2001) Constraining the Cosmological Constant from Large-Scale Redshift-

Space Clustering astro-ph/0105493


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, ∞


Lyman- ForestStructure along line-of-sight to QSOs

z

δzδθ

=12

δzδθ

=1

δzδθ

=2

Continuum

Fitting Systematic!

e−τ


The Ly- Alcock-Paczynski Forest TestMcDonald & 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

δzδθ

=12

δzδθ

=1

δzδθ

=2

0 1 2 3WM

- 1

0

1

2

3

WL

dzÄÄÄÄÄÄÄÄdq

=1, 2, 3, 4, 5, 6û z=2.35

NoBigBang


The Alcock-Paczynski Ly- Forest Test

z

δzδθ

=12

δzδθ

=1

δzδθ

=2

The quality of the AP test depends on the QSO separation• Too small• Just right• Too large


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


Ly- Forest Sensitivity

Cosmological Accuracy from SDSS QSOsMcDonald (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 .


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.

the alcock-paczynski probe of cosmology lyman- forest, lss, and cosmic consistency dark energy...

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