A&A 366, 1-6 (2001)
DOI: 10.1051/0004-6361:20000350
B. F. Roukema^{1} - G. A. Mamon^{2,3}
1 - Inter-University Centre for
Astronomy and Astrophysics,
Post Bag 4, Ganeshkhind, Pune, 411 007,
India
2 -
Institut d'Astrophysique de Paris
(CNRS UPR 341),
98bis Bd Arago, 75014 Paris, France
3 -
DAEC (CNRS UMR 8631),
Observatoire de Paris-Meudon, 5 place Jules Janssen, 92195 Meudon Cedex,
France
Received 8 September 2000 / Accepted 19 October 2000
Abstract
In the almost Friedmann-Lemaître
model of the Universe, the density parameter,
,
and the
cosmological constant,
,
measure curvature. Several linearly degenerate relations between these
two parameters have recently been measured.
Here, large scale structure
correlations at
Mpc are found in the comoving
three-dimensional separations of redshift
quasars. These function as a comoving standard rod
of length
Mpc.
A local maximum in the correlation function at
also appears to be significant.
By combining separate
radial and tangential standard ruler analyses,
the lifting of the
linear degeneracy within a single data set
is demonstrated for the first time.
Key words: cosmology: observations - cosmology: theory - distance scale - quasars: general - large-scale structure of Universe - reference systems
In standard cosmology
(Weinberg 1972), space is
a 3-manifold
(Schwarzschild 1900; Luminet & Roukema 1999)
of nearly constant curvature, i.e. space
is approximately locally homogeneous.
Geometric ways of measuring average curvature include the use of
phenomena of intrinsically fixed brightness or length scale,
i.e. of standard candles
(Perlmutter et al. 1999;
Riess et al. 1998)
and standard
rulers [e.g. Mo et al. 1992;
Broadhurst & Jaffe 1999;
Roukema & Mamon 2000;
note also the microwave background
angular statistical estimates
(Lange et al. 2000;
Balbi et al. 2000)
which can loosely speaking be thought of
as "theoretical'' standard rulers],
but have previously been found to lead to degeneracy in the
plane (e.g.
Lineweaver 1998).
However, inhomogeneities (perturbations) in density exist and
can be statistically represented
by a Fourier power spectrum, and are believed to gravitationally collapse
and form objects such as galaxies and clusters of galaxies.
Use of a characteristic feature of this spectrum at a scale
10
Mpc, the size of the largest bound structures,
should provide a comoving standard ruler
for constraining the local geometrical parameters
.
Figure 1: Projected comoving spatial distributions of the two quasar subsamples at redshifts , for . a) The right ascension subsample, for increasing from left to right. b) The declination subsample for increasing from left to right. The discrete redshifts published (Iovino et al. 1996) are converted to continuous values by uniform random offsets. The latter are used below. Differences in magnitude limits, hence different number densities, are visible in the subsample and are corrected for in both subsamples (cf. Table 1). What appears to be a cluster or a supercluster in b) at (+200,3150) Mpc is located at ( ) | |
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Many observations of both galaxies and superclusters of galaxies indicate that the maximum in the power spectrum is peaked at a wavenumber , where Mpc (Broadhurst et al. 1990; Gaztañaga & Baugh 1998; Einasto et al. 1997b; Deng et al. 1996) (comoving length units). Since this standard ruler should be valid independently of orientation, the different degeneracies implied in the radial and tangential applications of the ruler should enable lifting of the degeneracy within a single data set, providing a potentially more powerful ruler than previous standard rulers or standard candles.
It should be noted that while the existence of a broad maximum in the power spectrum is uncontroversial, not all observational analyses agree on whether or not there is a sharp feature in the power spectrum in this region in addition to the broad maximum, and there is not yet any clear agreement on the characteristic scale of the broad maximum. For example, on one hand, Einasto et al.'s (1997b) analysis of superclusters suggests a sharp peak at Mpc^{-1}. But, on the other hand, while in the low redshift IRAS PSCz (point source catalogue redshift) survey (Sutherland et al. 1999), there is, at least, a broad maximum at around 0.02Mpc^{-1} Mpc^{-1} (Fig. 1 of Sutherland et al. 1999), i.e. 320 Mpc Mpc, there is no obvious sharp feature in this region. Nevertheless, there is a significant sharp peak (see Fig. 1 and comment in Sect. 6 of Sutherland et al. 1999) which lies at 0.07Mpc^{-1} Mpc^{-1}, i.e. 90 Mpc Mpc in the PSCz.
Possible reasons why the Mpc feature found by other authors, if real, might have been missed in the Sutherland et al. (1999) analysis include
Physics which could potentially be investigated in order to explain the feature at includes acoustic oscillations in the baryon-photon fluid before last scattering, in high baryon density models (Eisenstein 1998; Meiksin et al. 1998; Peebles 1999), and features from Planck epoch physics which transfer to oscillations in the post-inflation power spectrum, for weakly coupled scalar field driven inflationary models (Martin & Brandenberger ). At high redshift, the Mpc feature ["distance'' means comoving proper distance (Weinberg 1972) throughout this Paper] has been detected among quasars (Roukema & Mamon 2000; Deng et al. 1994) and Lyman-break galaxies (Broadhurst & Jaffe 1999). Most applications of standard candles or standard rulers exploit either the radial redshift-distance relation (Broadhurst & Jaffe 1999) or the tangential redshift-distance-angle relation (Perlmutter et al. 1999; Riess et al. 1998; Roukema & Mamon 2000; Lange et al. 2000; Balbi et al. 2000), but not both simultaneously.
Alcock & Paczyñski (1979) suggested the idea of using both constraints simultaneously, and suggested applying it at quasi-linear or non-linear scales, i.e. Mpc, but did not discuss how to lift the degeneracy in the two curvature parameters which remains after using the local isotropy constraint, though they did suggest a theoretical method for separating out some of the peculiar velocity effects which are important at these small scales. Phillipps (1994), Matsubara & Suto (1996), Ballinger et al. (1996) and Popowski et al. (1998) followed up this idea, demonstrating specific formulae and calculations regarding quasar pairs and the two-point auto-correlation functions of galaxies and quasars, including separation of local isotropy ("sphericity'') and some of the peculiar velocity effects.
However, by using a standard ruler in the linear regime, i.e. by using a feature at Mpc, peculiar velocity effects become negligible, and the inability of this scale to evolve in a Hubble time provides an additional constraint in the plane. For the Mpc auto-correlation function, the peculiar velocity effects are certainly important, and evolution in the length scale must be contended with, for example by model-dependent assumptions.
N | |||||||
"Right ascension () subsample'' | |||||||
-42.0 | -37.5 | 500 | |||||
"Declination () subsample'' | |||||||
-42.0 | -28.0 | -37.5 | -32.5 | 453 |
Figure 2: Spatial two-point auto-correlation function , for separations r in comoving units and ( ). The four angular/redshift subsamples are shown as dashed ( ), dashed-dotted ( ), dotted ( ) and dashed-triple-dotted ( ) curves. The mean and the standard error in the mean are shown by the thick and thin solid lines respectively. The correlation functions are calculated in three-dimensional curved space via where DD, DR and RR indicate numbers of data-data, data-random and random-random quasar pairs respectively (Landy & Szalay 1993), and n=20 times more random points than data points are used. The random catalogues use (i) uniform probability distributions in the two angular directions (Table 1), and (ii) random permutations (``z scrambles'', IIIb in Osmer 1981) of the observational set of redshifts, to avoid biases from redshift selection effects (Scott 1991). Bin size is 5 Mpc and is smoothed by a Gaussian with Mpc. The low values of at Mpc are related to redshift roundoff error | |
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Figure 3: Correlation function , as for Fig. 2, but for a hyperbolic universe above, and a zero cosmological constant, flat universe below | |
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Figure 4: Confidence intervals for rejecting the presence of a peak at for various hypotheses on . Rejection levels are (white), (hatched), (light cross-hatched) and (heavy cross-hatched). The contour for testing a peak at test is shown in bold. For each pair , the peak position is estimated as the value for which is maximum in Mpc. (For the peak, Mpc is searched). The measurement uncertainty in the peak is estimated as (for f=0.5,1). The probability of finding close to assumes Gaussian errors, i.e. , where | |
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In a previous analysis (Roukema & Mamon 2000) of a deep, dense, homogeneous quasar survey (Iovino et al. 1996), only the tangential relation was used, to ensure that observational selection effects well known to cause non-cosmological periodicities in redshifts (Scott 1991) could not bias the result. In the present analysis of the high grade quasar candidate catalogue (Table 1, Fig. 1), the technique of "redshift scrambling'' (see Fig. 2 caption) is used to enable use of three-dimensional information in a way that avoids redshift selection effects. Since the redshifts used in the random and observational catalogues consist of exactly the same set of numbers, any redshift selection effects, which are independent of angle, should statistically cancel out (Osmer 1981) in calculation of the correlation function (Groth & Peebles 1977). Since some real signal could also cancel out, in principle, this implies a conservative estimate of , i.e. a lower limit to .
Figures 2 and 3 show that, for reasonable values of , a local maximum in the correlation function consistent with Mpc is clearly present. By contrast, an universe would require this local maximum to occur at Mpc, in contradiction with the low redshift estimates of . A correlation function consistent with the standard (Groth & Peebles 1977) galaxy-galaxy correlation function Mpc)^{-1.8} is also present for Mpc.
What is the significance
of the
peak? This depends on where the zero level of
correlation lies. In correlation function estimates where both
sample and correlation are small, the problem of only
having a finite volume often requires a correction known as
the integral constraint
(Groth & Peebles 1977),
which most often increases
the precorrected values of .
Making an
integral constraint correction usually requires assumptions on the
intrinsic shape of .
To avoid these assumptions,
it is more prudent just to quantify
the peak as a local maximum
(Deng et al. 1994; Roukema & Mamon 2000).
For a maximum at
consistent with a peak at
,
where f=1, define
and
as
the maximum value and the first minima below and above
respectively, and
take the maximum value of
for
,
where
Mpc. Then,
0.025 | 0.01 | 0.335 | < 0.005 | 0.005 | 0.005 | < 0.005 |
Figure 4 shows that an automatic search for this
peak, using a simple and robust method, i.e. using the value of
r for which
is maximum over a very large interval in r,
yields an approximately linear confidence band in the
plane. Since this band is consistent with
kinematical
(Carlberg et al. 1997;
Mamon 1993)
and baryonic fraction
(White et al. 1993;
Henriksen & Mamon 1994)
constraints for clusters and groups of galaxies,
though to slightly higher
values than were found
in the purely tangential analysis of the present survey
(Roukema & Mamon 2000), the
coincidence would be surprising if it were due to noise or
systematic effects.
Moreover, what appear to be peaks at
and near
are
present, though to lower significance, with
(S/N)_{0.5}=2.3 and
(S/N)_{2}=1.2 respectively,
for
(0.4, 0.6).
Could any of the three peaks be induced by noise which has
common statistical properties among all the four subsets,
either due to shot noise or
selection effects? Redshift selection effects
have been removed by the use of z-scrambling. Angular selection
effects may be present at a small level
(see Sect. 3.4 of
Roukema & Mamon 2000),
but are more likely to decrease the amplitude of any signal rather
than introduce false correlations which mimic the signal found.
Moreover, the convergence of the
separate tangential and radial analyses below (Sect. 2.2)
suggest that the effects of angular selection
are weak.
To test the properties of shot noise,
random simulations were performed as before but substituted
for the data. The probabilities that maxima can occur as close to
and of at least the same signal-to-noise
ratio as the observed values can be defined
Figure 5: Partial lifting of the degeneracy. Confidence intervals are as for Fig. 4 for a) the radial , b) tangential and c) combined constraints. d) A simple model, for which the contours for radial (straight contours) and tangential (curved, nearly vertical contours) constraints showing near linear degeneracy are shaded as before (hatched regions); and the combined contour is the half-ellipse-like shape nested between these (heavy contour). a-c) The radial and tangential tests are performed for the peak as in Fig. 4, except that only pairs oriented within ^ of the radial and tangential directions (respectively) are included in calculation of . Both angular subsamples over are used. The radial and tangential tests are assumed to be statistically independent. d) For the model, the redshift interval and the angle corresponding to , assuming that and that z=2.1, are calculated. For each pair , the radial and tangential distance intervals implied by and are calculated, ignoring the initial assumption about and . These are treated as two independent "experiments'', and Gaussian probabilities of observing these values, given , and are calculated as before. The combined rejection is | |
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Figure 4 shows that the confidence intervals for the and peaks are consistent.
A standard ruler should not depend on orientation. Can use
of both radial and tangential information lift the degeneracy of
constraints?
To illustrate
this, the full redshift interval
is used, but only pairs of objects within 30^
of either the
radial or tangential directions respectively are used.
For the geometry of this survey,
about 10% of pairs are radial and 60% of pairs are
tangential according to this criterion.
Figure 5
clearly shows, both observationally and theoretically,
the difference in the slopes of the radial and tangential constraints
at
.
A hyperbolic (
)
universe is suggested
by the 68% confidence limit, though
a flat universe with
is only rejected to
confidence, i.e. not significantly.
The partially lifted degeneracy can be represented (at 68% confidence) as
The confirmation of the peak and the partial lifting of the degeneracy show that ongoing and future large quasar surveys [in particular the 2 Degree Field Quasar Survey (Boyle et al. 2000) and the Sloan Digital Sky Survey quasar sample (e.g. Fan et al. 2000)] will have a much more powerful tool for local geometrical constraints than was previously thought. While local isotropy of the Mpc scale correlation function can in principle be used as a local geometrical constraint, a standard ruler at Mpc has the advantages (i) of being little affected by peculiar velocities, and (ii) of occurring well into the linear regime where evolution within a Hubble time is unlikely.
Moreover, the detection of the peak (cf. Fig. 6 of Tadros & Efstathiou 1996; Fig. 1 of Sutherland et al. 1999; Fig. 3 of Mo et al. 1992) implies that both peaks might either be signs of high baryon density (Eisenstein 1998; Meiksin et al. 1998; Peebles 1999) or of pre-inflationary physics (Martin & Brandenberger ), enabling constraints to be put on these. For increased confidence in this method, more precise low redshift constraints on large scale structure features near Mpc will be highly desirable. Results from the 2 Degree Field Galaxy Redshift Survey (e.g. Folkes et al. 1999), the Sloan Digital Sky Survey galaxy sample (York et al. 2000), and the 6 Degree Field galaxy survey (e.g. Mamon 1998) may help for these low redshift calibrations.
Acknowledgements
We thank Emmanuel Bertin, Stéphane Colombi, Georges Maignan, S. Sridhar and the referee, Pat Osmer, for useful comments. B.F.R. thanks the Institut d'Astrophysique de Paris, CNRS and DARC, Observatoire de Paris, for their hospitality, and acknowledges the support of la Société de Secours des Amis des Sciences. Data are available at http://cdsweb.u-strasbg.fr/cgi-bin/Cat?J/A+AS/119/265.