Statistical estimation of the shape

χ2(A , RV ,RE , V ,σ0,σ1)=∑

p

Sp [ (np−n(r p , z p))2

σ2(r p , z p , n (r p , z p))

+log σ (r p , z p , n(r p , z p))]Pixel mask

n0(r )=A+B ( rRV )

3

n( d , z )=n0(√( dRV )

2

+( zRE ,V )

2

)

Void profile model

Stretched void profile model

σ (d , z)=σ0 √1 Mpc/h

dσ1

if n(d , z )<nmax

otherwise

Error model

No correction Naive correction

Str

ech

ing

Redshift Redshift

x 1.16

Hubble diagram : effect of peculiar velocities on voids

Cosmic Voids As Standard Rulers for CosmologyGuilhem Lavaux1, Benjamin D. Wandelt2

(1) University of Waterloo / Perimeter Insititute ; (2) Université Pierre et Marie Curie (Paris 6) / Institut d'Astrophysique de Paris

FoM voids/BAO = 0.73 FoM voids/BAO ~ 60

Forecasts on constraints of Dark Energy physics

w0

w

a

w

a

w0

Sharp edge

Outer shell

d (Mpc/h)

z (M

pc/

h)

Cubic approximation (Dark matter)

(Angular coordinates)

(Red

shift

coo

rdin

ate)

Stacked void profile (N-body simulation)

Alcock-Paczynski test with voids

Density tracers (galaxies, particles)

Put in (r,z) coordinates

ZOBOV

Organize voids in a tree

Walk the tree from the root :- stop if volume constraints are met- optionally, stop when density constraints are met

Compute volume barycenter of selected voids

Extract spheres in (r,z), centred on barycenters

Extract a parallelepipedic volumemapped to a cube

Statistical analysis of the shape of stacked void

Alcock-Paczynski test

δ zδ r

=( DA (z )

z f ' k (χ (z))) H (z )

H (z=0)

Depends on m,

X, w

i

Tesselation : Zobov algorithm uses a Delaunay tesselation to find the local minima of the density field

Void tree structure : we store the found minima in a tree, and walk from the root.That allows separating the volume intoa non-overlapping set of voids.

Constraints on

cosmology

Bibliography

Ryden, B. S., 1995, ApJ, 452, 25

Alcock, C. & Paczynski, B., 1979, Nature, 281, 358

Lavaux, G. & Wandelt, B. D., MNRAS, 2010, 403, 1392

Neyrinck, M. C., 2008, MNRAS, 386, 2101

Lavaux, G. & Wandelt, B. D., ApJ submitted, arXiV : 1111.0345

AbstractWe show a purely geometrical method for probing the expansion history of the Universe from the observation of the shape of stacked voids in spectroscopic redshift surveys. This method is an Alcock-Pasczynski test based on the average sphericity of voids posited on the local isotropy of the Universe, which acts as rulers of unknown size. We describe the algorithm that we use to detect and stack voids in redshift shells on the light cone. We establish, and test on N-body simulation, a robust statistical model for estimating the average stretching of voids in redshift space. Finally, we discuss the constraining power on dark energy parameters in terms of the figure of merit of the Dark Energy Task Force. We estimate the figure of merit for SDSS, BOSS and EUCLID class surveys. For EUCLID, the figure of merit is an order of magnitude higher than Baryonic Acoustic Oscillation based methods.

Conclusion• We have developped an algorithm to use the average geometry of voids to constrain cosmology.• We have tested this method on simple mock light-cones using N-body simulation.• Some systematic bias is produced by peculiar velocities. We found that a naive correction is sufficient to recover the stretching at all redshifts.• We have produced some forecast of the expected constraints on Dark Energy using either the SDSS main galaxy sample or the future EUCLID galaxy spectroscopic sample.• This method seems to be a powerful competitor to Baryon Acoustic Oscillations, and should be studied in greater details.