ApJ in press Measuring q from the Distortion of Voids in Redshift Space 1 Barbara S Ryden Department of Astronomy The Ohio State University W th Ave Columbus OH ABSTRACT Because the transformation from distance to redshift is nonlinear maps in redshift space b ecome increasingly distorted as the redshift z b ecomes greater As noted by Alco ck Paczynski observed redshift distortions can b e used to estimate the deceleration parameter q If q is greater than voids 0 0 in redshift space will b e elongated along the line of sight In addition distant voids will have a greater volume in redshift space than nearby voids Accurate measurement of the volume and the axis ratio of voids as a function of their central redshift will provide an estimate of q 0 To test this metho d of estimating q I create a twodimensional toy universe 0 free of p eculiar velocities in which the galaxies are lo cated near the walls of Voronoi cells The galaxies are then mapp ed into redshift space adopting dierent values of q In redshift space I estimate the area and the axis ratio 0 astro-ph/9506028 5 Jun 95 of the voids by tting ellipses within the voids using an algorithm which maximizes the area of the empty ellipses and ensures that ellipses do not overlap The accuracy of the estimated values of q is limited by the intrinsic scatter 0 in the size and shap e of the voids In the toy universe distinguishing b etween a q universe and a q universe requires a survey which go es to 0 0 in redshift space Peculiar velocities will create an additional a depth z source of uncertainty for the values of q measured in Nb o dy simulations and 0 in the real universe Subject headings cosmology theory galaxies distances and redshifts largescale structure of universe National Science Foundation Young Investigator rydenmpsohiostateedu Introduction A quartercentury ago cosmology was characterized as the search for two numbers H and q Sandage Although the scop e of cosmology has broadened those same 0 0 two numbers are still b eing assiduously searched for I dont intend at the moment to plunge into the H fray However in this pap er I will describ e how q may b e determined 0 0 in principle by measuring the size and shap e of voids in redshift space The classical tests for q as describ ed by Sandage involve measuring the ux of 0 standard candles or the angular size of standard rulers The standard candles used to test for q are usually galaxies Accurately determining q therefore requires knowing how the 0 0 intrinsic luminosity and color of galaxies evolve with time Unfortunately for cosmologists galaxy evolution is a complex pro cess Uncertainties in the evolutionary mo dels lead to a large uncertainty in the derived values of q Among the ob jects used as standard rulers to 0 determine q are brightest cluster galaxies Sandage extended radio sources Miley 0 Kapahi Daly and clusters of galaxies Hickson Adams Bruzual Spinrad Once again the diculty of determining the intrinsic evolution results in a large uncertainty in the value of q which is determined 0 It is desirable therefore to measure cosmological parameters by using metho d which are uncontaminated by evolutionary eects One such metho d was prop osed by Alco ck Paczynski who prop osed measuring the width and depth in redshift space of ob jects which are intrinsically spherical or nearly so in real space The distortion of the spheres in redshift space would provide an estimate of q and the cosmological constant 0 At the time of publication Alco ck and Paczynski had no data to which they could apply their test so their pap er remained an undercited curiosity Recently however interest in their geometrical metho d has revived Phillips has prop osed using quasar pairs assumed to b e randomly oriented in space as a substitute for the intrinsically spherical ob jects of Alco ck and Paczynski Placing interesting limits on q using quasar pairs will 0 require observations of approximately pairs at redshifts z In this pap er I will apply the metho d of Alco ck Paczynski to galaxies at relatively small redshift z I will describ e how the size as well as the shap e of voids in redshift maps can b e used to estimate q In section I describ e how measurements in redshift space can b e used 0 in principle to measure q In section I explain a practical metho d for measuring the 0 shap es of voids and apply it to simple toy universes Section contains a discussion of the practicality of applying the void distortion test to real redshift surveys Theoretical Exp ectations Consider the idealized case of a spherical void which is surrounded by a thin wall of galaxies At a time t the void wall has a radius r t and the galaxies within the wall are v moving outward with a velocity u t At some later time t an observer lo cated outside v 0 the void collects light that was emitted from the void walls at the time t plus or minus the light travel time across the void The redshift of a galaxy on the near side of the void is z z and the redshift of a galaxy on the far side is z z where z is the redshift of the void center and u t v z z c The angular radius of the void as seen on the sky is H r t z 0 v c y z where H is the Hubble constant at time t and y z is the dimensionless angular size 0 0 distance Peebles In a spatially at universe the angular size distance is Z z dz y z H 0 H z 0 If the void is plotted in redshift space the ratio of its depth to its width is z u y z v e z v z H r z 0 v In general the function e is not equal to one even for a void which is intrinsically spherical v and which is simply expanding along with the Hubble ow If the galaxies in the void wall are expanding along with the Hubble ow then u H r However voids tend to expand more rapidly than the Hubble ow Thus for a v v spherically expanding void u t H tr t F t v v with the excess expansion factor F b eing greater than zero for voids expanding more rapidly than the Hubble expansion In a spatially at matterdominated universe an isolated spherical void of nite radius evolves into a selfsimilar state with F if the void is comp ensated that is if the net mass decit is equal to zero and F if the void is uncomp ensated Bertschinger If the initial underdensity of the void has a p owerlaw form / r then F when Fillmore Goldreich When the initial density prole has then the void wall is a density wave and matter moves outward with F Ryden If the universe is not at then F is a function of the density parameter In an op en universe Regos and Geller nd 06 that comp ensated voids have F Thus an isolated void when plotted in redshift space will b e stretched in the radial direction as a result of its p eculiar expansion velocity in addition to the distortion which results from cosmological factors The observed distortion of a void will b e H t y z e t F t v H z 0 with the term in parentheses representing the cosmological distortion and the term in square brackets representing the distortion due to the p eculiar velocity of the voids expansion In order to determine cosmological parameters from the observed distortion of voids in redshift space I must somehow distinguish b etween the cosmological distortion and the p eculiar distortion Observed voids are not isolated structures The distribution of galaxies in redshift space is bubbly galaxies are typically lo cated in relatively narrow walls b etween voids which are nearly spacelling The average void size when measured in comoving units can only increase by the merger of two adjoining voids to form a single void or by the expansion of one void at the exp ense of a neighboring void Consider two adjacent voids with radii r and r If r r then the wall b etween the voids will have no p eculiar 1 2 1 2 velocity in the direction p erp endicular to its surface if r r the void wall will b e pushed 1 2 in the direction of the smaller void with an excess velocity Regos Geller 06 F r r 2 1 compared to the Hubble expansion velocity H r In addition to the velocity p erp endicular 1 to the void wall the galaxies within the wall tend to ow outward within the plane of the wall The pair of neighboring voids is thus converted into a single void as the intervening wall breaks up The velocities tangential to the void wall have a maximum excess velocity F when compared to the Hubble expansion velocity H r Dubinski et al 1 In a hierarchical scenario the universe at any given time is lled with voids of a characteristic size R Dubinski et al Since all voids are roughly the same size the v p eculiar velocities in the direction p erp endicular to void walls as found from equation should b e small even if is close to one The p eculiar velocities in the direction parallel to the void walls may have values of F as large as p ercent However if the evolution of the hierarchical universe is approximately selfsimilar then the average value of F should remain constant with redshift The average distortion of voids in redshift space due to p eculiar velocities will b e constant with redshift the distortion of voids due to cosmological eects will increase with increasing redshift Observationally there is no tendency for nearby voids to show distortions resulting from p eculiar velocities The analysis by Slezak de Lapparent and Bijaoui of the rst CfA redshift slice picks out voids whose centers are at a redshift z This small sample of nearby voids shows no tendency to b e elongated along the line of sight in