Issue 
A&A
Volume 533, September 2011



Article Number  A58  
Number of page(s)  2  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201117559  
Published online  26 August 2011 
Friedmannfree limits on spatial curvature
CEA
Centre de Saclay, Irfu/SPP,
91191
GifsurYvette,
France
email: rich@hep.saclay.cea.fr
Received: 24 June 2011
Accepted: 30 July 2011
We discuss limits on cosmological spatial curvature that can be derived without imposing the geometrydensity relation required by the Friedmann equation. In particular, studies of the expansion history using stellar evolution in passive galaxies imply a curvature radius greater than the Hubble distance.
Key words: cosmological parameters
© ESO, 2011
The standard ΛCDM cosmological model has been deduced from astronomical data that are generally interpreted within the FriedmannLemaîtreRobertsonWalker framework for a homogeneous universe. The framework uses the RobertsonWalker metric with the proper time as a function of comoving coordinates given by (1)The dimensionless scale parameter a(t) reflects the universal expansion relative to the present epoch, t_{0}, a(t_{0}) = 1. The spatial curvature parameter, K, can be positive or negative and is related to the curvature radius by . The metric allows one to calculate trajectories of test particles (e.g. photons) in terms of a(t) and K which are governed by the Friedmann equation: (2)for an energy density ρ(t).
Cosmological data analyzed in this framework imply a nearly flat (r_{c} ≫ c/H_{0}) model where the universal expansion is accelerating (). Both of these conclusions concern parameters of the RobertsonWalker metric so it is interesting to see to what extent they can be derived without assuming that a(t) is governed by the Friedmann equation, i.e., assuming homogeneity but not any particular gravitational dynamics. This is especially true given that the observed acceleration has encouraged speculation that gravity might be modified at cosmological scales.
By measuring directly the deceleration parameter, , data on moderateredshift type Ia supernovae (SNIae) imply acceleration independent of the Friedmann equation (Shapiro & Turner 2006). Sutherland (2011) has recently proposed a Friedmannfree test for acceleration using baryon acoustic oscillation (BAO) effects in the matter correlation function.
Studying spatial curvature without using the Friedmann equation is a bit trickier. The very strong published limits on r_{c} use the Friedmann equation evaluated at the present epoch which yields the geometrydensity relation: (3)where H_{0} is the present value of the expansion rate and where Ω_{T} is the present energy density in units of the critical density . Cosmic microwave background (CMB) data combined with BAO and SNIa data imply −0.0178 < (1 − Ω_{T}) < 0.0063 (Komatsu et al. 2011) which gives (4)
In the cosmology of a homogeneous universe, the spatial curvature determines the relation between the present distance, d(z), to an object of redshift z and the luminosity and angular distances, d_{L}(z) and d_{A}(z). These two distances determine the flux, F, from objects of luminosity, L, and the angular size, Δθ, of an object of size R: In the absence of curvature, d_{L} and d_{A} are both proportional to d(z), with factors of proportionality (1 + z) for d_{L} and (1 + z)^{1} for d_{A}. For universes with K > 0 (K < 0), d_{L} and d_{A} are less than (greater than) what would be expected from proportionality.
It is not easy to test this fundamental prediction. The basic problem is that while d_{A} and d_{L} can be determined directly by using standard candles and rulers, d(z) can only be calculated using knowledge of the expansion rate: (5)where the redshift is given by z(t) + 1 = 1/a(t). The limits of integration are given by the times of signal emission, t_{1}, and reception, t_{0}. The luminosity and angular distances depend on r_{c}: (6)where S_{1}(x) = sin(x) and S_{1}(x) = sinh(x) for K > 0 and K < 0. Equation (6) defines the relation between d(z) and d_{L,A}(z) via the curvature radius r_{c}. If d(z) were known, it would allow a determination of r_{c} and a verification that it takes the value (3).
If a(t) and/or H(t) are not directly measured, d(z) can only be calculated by using the Friedmann equation in Eq. (5). This imposes the value of r_{c} given by (3). As such, the fundamental relationship (6) cannot be tested in this way. Rather, the luminosity and angular distances are used to determine cosmological parameters (Ω_{M},Ω_{Λ},Ω_{T}) that can then be used to calculate the distance d(z).
To avoid use of the Friedmann equation, one needs an independent measurement of H(z) to be injected into Eq. (5). Probably the best possibility is to use the radial BAO feature in the matter correlation function which gives directly H(z). Clarkson et al. (2008) noted that this could directly measure spatial curvature. Note that while the length of the BAO standard ruler depends on the Friedmann equation in the prerecombination universe, the measurement of H(z) depends only on it being a comoving ruler, i.e. that galaxies having nearly fixed comoving coordinates. This is a nontrivial assumption but one that does not depend explicitly on the Friedmann equation.
In the absence of BAO measurements of H(z), we note that if galaxies were equipped with clocks that we could see and read, measuring distances would be trivial since an ensemble of such galaxies at different redshifts would yield z(t) from redshift and time measurements (Jimenez & Loeb 2002). Distances could then be calculated using Eq. (5) without imposing the Friedmann equation.
Stellar evolution in passive galaxies (those with negligible star formation) can be used as such a clock. The first steps in their use for cosmology have been performed by Simon et al. (2005), Figueroa et al. (2008) and Moresco et al. (2011). They derive expansion histories that are consistent with the standard flat ΛCDM cosmology (Ω_{Λ} = 0.73 and Ω_{M} = 0.27). The most precise results were obtained by Moresco et al. (2011) who observed the redshift evolution of the 4000 Åspectral break in SDSS elliptical galaxies in the range 0.15 < z < 0.30. Stellar population evolution models then gave the elapsed time between z = 0.15 and z = 0.3 necessary for the spectral evolution. Using H(z) = ż/(1 + z), this gives the expansion rate for 0.15 < z < 0.30. They extrapolated this rate to zero redshift using the standard cosmology and found a value H_{0} = 72.6 ± 2.9(stat) ± 2.3(syst) km s^{1} Mpc^{1}, that is in agreement with recent local measurements, H_{0} = (74.2 ± 3.6) km s^{1} Mpc^{1} (Riess et al. 2009). From our point of view, this means that at the 5% level the expansion rate in the range 0 < z < 0.3 follows the flat ΛCDM prediction.
Since the measured expansion rate follows the flat ΛCDM prediction, Eq. (5) implies that the distances d(z) to objects in this redshift range agree with the flat ΛCDM values. Furthermore, the mean luminosity distances in this redshift range from SNIa (Kessler et al. 2009) agree with those of the standard flat ΛCDM model to a precision of better than 5%. The agreement of both d(z) and d_{L}(z) with the same flat model implies that spatial curvature effects must be small at z = 0.3.
To quantify the limit on r_{c}, we invert (6), giving d(z) as a function of d_{L}(z), and then take the derivative with respect to z. Using d′(z) = 1/H(z) and the fact that d_{L}(z) is well described by a flat ΛCDM model, one finds (7)where H_{flat} and d_{flat} are the expansion rate and distance for the flat ΛCDM model that gives the observed luminosity distance. For the measurement of H(z ~ 0.22) of Moresco et al. (2011), if we adopt a conservative limit (8)we find (9) (10)The work of Figueroa et al. (2008) provides H(z) up to z ~ 1.5, but at a precision of only ~20%. Adopting (11)we find (12) (13)Here again, we use the fact that SNIa for 0.3 < z < 1.5 (Guy et al. 2010; Riess et al. 2007) give luminosity distances that are in agreement with flat ΛCDM. Baryon acoustic oscillations measurements (Eisenstein et al. 2005; Percival et al. 2010; Blake et al. 2011) imply the same for the distance combination d_{A}(z)^{2}c/H(z) (0.2 < z < 0.6 ).
The limits on r_{c} obtained here are rather modest, implying only Ω_{T} < 1.6 if one assumes the validity of the Friedmann equation. It will be possible to confirm and improve the limits in the future with BAO H(z) measurements (White et al. 2011; Slosar et al. 2011). We note that BAO has the advantage of not depending on stellar modeling. A particularly interesting challenge would be to perform a Friedmannfree CMB analysis with the hope of extending the limits to z = 1070. Unfortunately, unless our ideas of gravity are wrong at cosmological scales and the Friedmann equation does not apply, the present limits on Ω_{T} mean that it will be very difficult to directly see cosmological curvature with these methods.
Acknowledgments
I thank Raul Jimenez, JeanBaptiste Melin, Will Sutherland, and JeanPhilippe Uzan for stimulating discussions. I thank the referee for comments that improved the paper.
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