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Radio source angular sizes on VLBI scales: an application to a classical cosmological test S. Frey ¨ FOMI Satellite Geodetic Observatory, P.O. Box 546, H– 1373 Budapest, Hungary

Abstract A novel approach to define milliarcsecond scale angular sizes of active galactic nuclei using VLBI visibility data is presented. The ‘‘angular size–redshift’’ relation for a sample of nearly 250 AGN is discussed. The analysis is based on 5 GHz VLBI observations made in the VSOP pre-launch survey and various imaging observations of extremely high redshift quasars. 1999 Elsevier Science B.V. All rights reserved. PACS: 98.54.Aj; 98.54.Cm; 98.70.Dk; 98.80.Es Keywords: Radio continuum: galaxies; Galaxies: quasars; Cosmology: observations

1. Introduction As was first pointed out by Kellermann (1993), the ‘‘angular size–redshift’’ relation for compact radio sources observed with VLBI shows a nonEuclidean behaviour. The apparent flatness of the angular size dependence from a redshift of z | 0.5 out to z | 4 can be interpreted as a signature of a universe with q0 . 0. This cosmological test for radio-loud active galactic nuclei (AGN) was first applied to estimate numerically the cosmological deceleration parameter (q0 ) by Gurvits (1994). The first results stimulated considerable discussion in the literature over the past few years. See Gurvits et al. (1999) for a recent review of developments in this field.

2. Characteristic angular size definition One of the important questions in the study of the ‘‘angular size–redshift’’ relation is how to assign characteristic angular-size values to the compact

radio sources observed with VLBI. Kellermann (1993) used 5 GHz contour maps to measure the angular distance between the strongest component (core) and the most distant component with peak brightness exceeding 2% of the peak brightness of the core. A similar definition was used by Wilkinson et al. (1998) with a 1% threshold. A sample of 330 currently published 5 GHz contour maps of AGN with known redshift was studied by Gurvits et al. (1999). The use of this approach is somewhat constrained by the limited number of sufficiently high dynamic range imaging observations available. Simulations by Dabrowsky et al. (1995) suggest that at least several hundred sources are needed to estimate the value of q0 usefully. Two-point VLBI visibility data of 337 sources were used by Gurvits (1994) to determine source characteristic angular sizes. This approach is extended here to analyse visibility data directly, as suggested by Gurvits et al. (1998), to handle more structural information (i.e. more visibility points per source). Apart from the increase in the number of objects in the sample, reduction of possible luminosity and

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frequency-dependent selection effects could further strengthen the statistical confidence of the parameter estimations. On the one hand, this can be done by applying selection criteria in the definition of AGN sample. Sources in relatively narrow luminosity range may be selected and multi-frequency VLBI observations may be used to match approximately the emitted frequency nem 5 nobs (1 1 z) for sources at different redshifts in contrast to the samples with fixed observed frequency (nobs ) studied so far. Selection effects could also be reduced by studying regions that are closer to the ‘‘central engine’’ of AGN. These ‘‘central engines’’ are thought to be controlled by only a few physical parameters (Gurvits et al., 1999) and are therefore good candidates for cosmological ‘‘standard’’ objects. Our goal is to find an adequate angular size definition which characterizes the compact cores of AGN and allows us to combine VLBI observations made at different (observed) frequencies and within a wide range of imaging sensitivity. This can be achieved by fitting a sky brightness distribution model to the visibility data (Pearson, 1995). The circular Gaussian model widely used in non-imaging VLBI data analysis has two parameters to adjust: the total flux density of the source (S0 ) and the FWHM intensity (u ). The latter can be considered as a measure of the source angular diameter. The Fourier transform of the model centered at the origin of the sky coordinate system in the (u,v) plane is

S

D

2 (pur )2 Sc ( r ) 5 S0 exp ]]] , 4 ln 2

(1)

where Sc is the correlated flux density or visibility amplitude and r is the projected baseline length or (u,v) distance in the of wavelengths. The measured visibility amplitude values are given as a function of projected baseline length: Sci ( ri ) (1 # i # n), where n is the number of data points. Finding a maximum likelihood estimate of the model parameters S0 and u is performed by minimizing the quantity S 2 S (r ) OS]]]] D, s n

x2 5

i 51

2

ci

c

i

(2)

i

where s 2i is the quadrature sum of the uncertainty in the ith correlated flux density value Sci and the

standard deviation of the correlated flux densities in the corresponding (u,v) distance range. Due to the relative simplicity of the circular Gaussian model, it does not describe the brightness distribution of all radio sources equally well. The goodness of fit can be assessed based on the value of the reduced chi-square x 2 /(n 2 2). In practice, less than about 20% of the sources had to be omitted from the subsequent analysis because the complicated source structure could not be adequately fitted with such a model. In the framework of beaming models, if this introduces a selection effect, it tends to confine the sample to sources with beam orientations closer to the line of sight. This reduces possible source orientation bias.

3. Application to a sample: results We used a sub-sample from the 5 GHz VSOP pre-launch VLBA survey (Edwards & Fomalont, 1998) that consists of sources with known redshifts observed with sufficiently high resolution to study the ‘‘angular size–redshift’’ dependence and to estimate the value of q0 . The visibility data of 246 sources were supplemented with imaging observations of a further 13 extremely distant (z . 3) quasars (Frey et al., 1997; Paragi et al., 1999). Source angular sizes were determined using the method described above. After the model fitting, the remaining 208 sources were grouped into 14 nearly equally populated redshift bins. The redshift distribution of the sources is shown in Fig. 1. We performed a multi-parameter regression analysis (Gurvits, 1994; Gurvits et al., 1999; Frey, 1999) capable of estimating the value of q0 as well as the characteristic linear size (lh) and the luminosity- and redshift-dependence ( b and n) of the linear size in the framework of Friedmann world models (with zero cosmological constant), assuming

u ~lh L b (1 1 z)n D 21 (z,q0 ),

(3)

where D is the luminosity distance. (The Hubble constant is H0 5 100h km s 21 Mpc 21 .) The best-fit model in terms of reduced x 2 obtained after investigating a large set of trial parameters is lh 5 3.6360.43 pc and q0 5 0.3860.71, with b 5 0.3 and

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Fig. 1. Redshift distribution of the source sample used for the regression analysis. Open and cross-hatched columns indicate the distribution of sources from the VSOP pre-launch survey and supplementary z . 3 quasars (Frey et al., 1997; Paragi et al., 1999) respectively.

n 5 2 0.6 kept fixed (Fig. 3). Note that the characteristic linear size of the ‘‘standard rod’’ is about an order of magnitude smaller than that involved in the analysis of contour maps by Gurvits et al. (1999). This is due to the fact that our source-size definition deals with the compact VLBI cores rather than the

more extended jet structures, and thus characterises regions that are physically closer to the ‘‘central engines’’ of AGN. The analysis described above is intended to be a first demonstration of this approach. Although this specific study does not use all the advantages of this

Fig. 2. Luminosity Lh 2 at 5 GHz (rest frame) of the sources in the sample assuming zero spectral index. (H0 5 100h km s 21 Mpc 21 ; q0 5 0.5 is used for making this plot only.)

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source angular size obtained in this way is characteristic of the innermost regions of AGNs accessible with the current angular resolution of VLBI. This size definition allows us to match more straightforwardly the redshift-dependent angular resolution to an appropriate linear resolution (i.e. intrinsic source physical size) over the whole redshift range in the sample. The feasibility of the approach was demonstrated using 5 GHz VLBA data from the VSOP pre-launch survey and other VLBI imaging observations of extremely distant quasars. A regression analysis led to the best fitting model with an estimate of q0 5 0.3860.71. This is in qualitative agreement with estimates obtained with different angular size definitions using samples of comparable size. This indicates the potential of this or a similar method in analysing the ‘‘angular size–redshift’’ relation of considerably larger and more carefully selected samples of AGN in the near future. Fig. 3. Median characteristic angular size versus redshift. The 208 sources are grouped into 14 nearly equally populated redshift bins. The solid curve corresponds to the best fit model assuming hypothetical ‘‘standard rods’’ with constant luminosity. Note that the apparent large deviation of the curve from the data point in the lowest redshift bin is due to the dependence of the linear size on the luminosity. Nearby sources in our essentially flux-density limited sample have relatively low luminosity (see Fig. 2).

method for determining angular sizes, the result obtained is in good agreement with recent estimates using different VLBI samples and different definitions of angular size (Gurvits, 1994; Gurvits et al., 1999), or even using double-lobed radio quasars as ‘‘standard objects’’ on arcsecond scales (Buchalter et al., 1998).

4. Conclusion A definition of a milliarcsecond- and submilliarcsecond-scale characteristic angular size of AGN was described based on fitting a circular Gaussian brightness distribution to source visibility amplitudes. This approach is less sensitive to limitations in image dynamic range than is the use of contour maps. It allows us to combine observations having a wide range of imaging sensitivity as well as observations made at different frequencies. The measure of

Acknowledgements This research was supported in part by the EU under contract No. CGHECT 920011, the Netherlands Organization for Scientific Research and the Hungarian Space Office. The author thanks L. Gurvits for extensive consultation and advice, and E. Fomalont, P. Edwards, Z. Paragi, W. Scott and L. Gurvits for their work in survey data reduction.

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