Probing the curvature of the Universe from supernova measurement

We study the possibility to probe the spatial geometry of the Universe by supernova measurement of the cubic correction to the luminosity distance. We illustrate with an accelerating universe model with infinite-volume extra dimensions, for which the 1$\sigma$ level supernova results indicate that the Universe is closed.

The precision measurements of the Wilkinson Microwave Anisotropy Probe (WMAP) have provided high resolution Cosmic Microwave Background (CMB) data [1,2] and elevated cosmology to a new maturity. Among interesting conclusions that have been reached from these data, the WMAP results indicate that while flatness of the Universe is confirmed to a spectacular precision on all but the largest scales [1], a closed universe with positively curved space is marginally preferred [3,4,5,6,7]. This tendency of preferring closed universe is not restricted to the WMAP data, it appeared in a suite of CMB experiments before [8,9,10].
The improved precision from WMAP provides further confidence.
In addition to CMB, recently it was argued that the cubic correction to the Hubble law measured with high-redshift supernovae is another cosmological measurement that probes directly the spatial curvature [11]. This is the first non-CMB probe of the spatial geometry, which can provide a cross-check to the result got by CMB. In a toy model, it was already found that a curvature radius is larger than the Hubble distance [11].
Our Universe is accelerating rather than decelerating. This may be regarded as the evidence for a nonzero but very small cosmological constant (see [12] for a review and related references in [13]). Another possibility is that the phenomenon of accelerated expansion is caused by a breakdown of the standard Friedmann equation due to the extra-dimensional contribution [14,15,16,17]. Studies on this possibility can also be found in [18]. In this work we will consider the accelerated universe model resulted from the gravitational leakage into extra dimensions [16]. We will attempt to extract information from the full redshift data to test the spatial geometry.
Consider the accelerating universe described by the model with infinite-volume extra dimensions [16], the Friedmann equation is expressed as where ρ is the total cosmic fluid energy density and r c is the crossover scale. Eq. (1) can also be recasted in terms of the redshift as where Ω rc = 1 , Ω M is the non-relativistic matter density. The conservation for energymomentum tensor of the cosmic fluid is still described bẏ Using definitions q 0 = −ä ȧ a 2 | 0 , j 0 = ... a a 2 a 3 | 0 with dot denoting the differentiation with respect to time t for the deceleration parameter and the "jerk", respectively, we have directly from where the normalization of (2) at the present epoch has been employed. With (6), q 0 and j 0 are only determined by Ω M and Ω k 0 .
The physically reasonable cosmic model has the following requirements [19]: (1) the total density is currently not increasing as a function of time; (2) for causality and stability, the present sound speed c s of the total system satisfies 0 ≤ c 2 s ≤ 1. Employing (1) and (3), the variation of the total cosmic fluid energy density and the sound speed of the total cosmic fluid at the present epoch arė The first requirement implies Employing (6) and the fact that |Ω k0 | ≤ 0.1 as a consequence of CMB data, the above requirement reduces to Using (4), we see that Eq. (10) can obviously be satisfied.
The second requirement can now be written in a simplified form as where . Substituting Eqs. (4) and (6) into the expression of f 1 , we find that j 0 = f 1 , which means that the sound speed of the total system in this model is exactly zero.
We now turn to determine the cosmological density parameters from the supernova (SN) Ia data compiled by Riess et al. [20]. The likelihood for the parameters Ω M and Ω k 0 can be obtained from a χ 2 statistics [20,21], where µ p = 5 log 10 (d L /Mpc) + 25 and µ o are distance modulus for the model and the observations, respectively. d L is the luminosity distance defined for the Friedmann-Robertson-Walker universe model as (2), (6), (12) and (13).
Using the contour Ω k 0 , Ω M values, we can get the corresponding q 0 , j 0 and f 1 as plotted in Fig. 2. Note that the contours shown here are from the gold sample SN Ia data compiled in [20].
Lines added in Fig.2  for Ω k 0 > 0. This corresponds to say that the data favors the closed universe almost at 2σ level.
To obtain tighter constraints on the parameter space, we also include constrains from combined WMAP data [1,2] and SN Ia data. We minimize where σ R is the uncertainty in R, the CMB shift parameter R ≡ Ω 1/2 M H 0 r 1 (z ls ) = 1.710 ± 0.137 [22] and z ls = 1089±1 [1,2]. The results are shown in Fig.3. The combined constraints give Ω M = 0.25 +0.05 −0.04 and Ω k0 = 0.01 +0.09 −0.08 . This shows that in the absence of positive spatial curvature, Ω M tends to take a smaller value. It implies that from the observed Ω M around 0.3, we should have the positive curvature. The corresponding crossover scale r c = 1.04H −1 0 from supernova data and r c = 1.34H −1 0 from combined CMB and SN Ia data. This constrained parameter is in good agreement with the result comes from lunar laser ranging experiments that monitor the moon's perihelion procession with a great accuracy [24].
In summary, we have probed the geometry of a specific model describing the accelerating universe by using the full redshift data in supernova measurements. To almost 2σ level, our result indicates that the universe is closed. This result is also favored by including WMAP data constraint, which agrees to a suite of CMB experiments. The result obtained is consistent with the interpretation from other models, e.g. the matter plus cosmological constant case, that the Riess et al. data show a tendency towards a closed universe. Of course it is too early to draw conclusions just on 2σ level data, and we expect that future supernova measurements can determine the spatial curvature precisely. helpful discussions with Prof. E. Abdalla.