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FIG. 1: Some representative forms for the curvature function, k(r), and the corresponding distance moduli, normalised to an 3 empty Milne cosmology, ∆dm. Void (a) is a Gaussian in r (with FWHM r0), void (b) is a k ∝ exp{−c|r| }, and void (c) is k ∝ {1 − | tanh(r)|}, all normalised to the curvature minimum k0. The three diagrams on top indicate the spatial distribution of energy density in the three voids at some sample time (solid line for void (a), dashed line for void (b) and dotted line for void (c)). The ascending thick solid line on the right hand plot corresponds to a universe containing dark energy only (de Sitter space), and the descending one to a flat universe containing only dust (an Einstein-de Sitter universe). of the center of the void [9]. There have also been some As the space-time evolves the energy density in the vicin- attempts at calculating predictions for CMB anisotropies ity of the curvature perturbation is then dispersed, and and large scale-structure [10, 11, 12], as well as the kine- a void forms. Observations of distant astro-physical ob- matic Sunyaev-Zeldovich effect [13]. jects in this space-time obey a distance-redshift relation

General Relativity allows a simple description of time- 2 dependent, spherical symmetric universes: the Lemaˆıtre- DL = (1+ z) rE a1(tE, rE ) (5) Tolman-Bondi (LTB) models [14, 15, 16], whose line- element is where rE ′ 2 2 (˙a1r) 2 2 a2(t, r)dr 2 2 2 1+ z = exp 2 dr , (6) ds = dt + 2 + a1(t, r)r dΩ , (3) Z0 √1 kr  − 1 k(r)r − − ′ and subscript E denotes the value of a quantity at the where a2 = (ra1) , and primes denote r derivatives . The moment the observed photon was emitted. This expres- old FRW scale factor, a, has now been replaced by two sion is modified from equation (2), allowing for the pos- new scale factors, a1 and a2, describing expansion in the sibility of apparent acceleration without dark energy. directions tangential and normal to the surfaces of spher- We find that the form of the void’s curvature profile ical symmetry. These new scale factors are functions of is of great importance for the observations made by as- time, t, and distance, r, from the centre of symmetry, tronomers at its centre. In Figure 1 we plot some simple and obey a generalization of the usual Friedman equa- curvature profiles, together with the corresponding dis- tion, (1), such that tance moduli as functions of redshift (distance modulus, 2 ∆dm, is defined as the observable magnitude of an as- a˙ 1 8πG k(r) trophysical object, minus the magnitude such an object = ρ˜ 2 . (4) a1  3 − a1 would have at the same redshift in an empty, homoge- neous Milne universe). It is clear from Figure 1 that 3 Hereρ ˜ = m(r)/a1, and is related to the physical energy for the void models there is a strong correlation between ′ density by ρ =ρ ˜+˜ρ ra1/3a2. The two free functions, k(r) k(r) and ∆dm; at low redshifts ∆dm(z) traces the shape and m(r), correspond to the curvature of space, and the of k(r). Hence, for a generic, smooth void ∆dm starts distribution of gravitating mass in that space. We choose off with near zero slope, where it is locally very similar initial conditions such that the curvature is asymptoti- to a Milne universe, it then increases at intermediate z, cally flat with a negative perturbation near the origin, and later drops off like an Einstein-de Sitter universe. 5 and so that the gravitational mass is evenly distributed. For ΛCDM, we have ∆dm q0z at low z, i.e. a ≃ − 2 3 non-zero slope. Thus, although one can always find a be distinguishable with future surveys. The best fitting void profile that will mimic ΛCDM [17], such a void will void is 71 7% underdense at its centre, and has a scale have a curvature profile that is strongly cusped, and non- corresponding± to 850 170h−1Mpc today. This is of the differentiable, at z = 0 [18] (see the dotted line in Figure order expected to produce± a feature in ∆dm on a scale 1 or [19]). Conversely, any generalized dark energy model of z 0.6, and large enough to avoid strong constraints capable of producing a flat ∆dm at low z would be re- from∼ galaxy surveys that extend to z 0.1. On the quired to change the equation of state extremely rapidly other hand, the best fitting ΛCDM model∼ contains 74 between z 0.5 and 0.1. We therefore have a definitive 4% dark energy, and fits the data slightly better with± way to distinguish≃ between a realistic smooth void model, ∆ ln E 2.7. Thus, while the current data marginally and ΛCDM. prefers| | ΛCDM, ≃ it is not yet able to distinguish between the two models decisively. One will, of course, be interested in results from other SNe compilations. Using the Riess gold data [24], with the MLCS2k2 light-curve fitter, we find our basic results do not change significantly, with a ΛCDM model still being marginally preferred. Thus our analysis does not appear to be substantially effected by the apparent sys- tematic error that led to the identification of the ‘Hubble Bubble’ anomaly [25]. The ‘Union’ data of [26] is the FIG. 2: The current best fit ΛCDM and Gaussian void mod- largest compilation of SNe fitted for with the more con- els as dashed and solid lines, with triangular and square servative SALT fitter, and for these 315 SNe (including data points, respectively. The data shown here is a compi- the ‘outliers’) we find the void model is marginally pre- lation of 115 low and high-z SNe from the SNLS, fitted using ferred over ΛCDM, with ∆ ln E 2.5 in favor of a void. SALT [20]. For illustrative purposes we have binned the re- It therefore appears neccessary| |≃ to obtain more data, in sults with 10 SNe per bin (except the last one, which contains order to be able to decisively distinguish the two models. 5). Due to the uncertainty in the ‘nuisance’ parameters of the calibration the data points and the error-bars shift when fit- ting for different models.

We will now compare the smooth void model to the first-year SNe Legacy Survey (SNLS) data, consisting of 115 SNe [20] calibrated with the SALT light-curve fitter[21], and contrast it with ΛCDM. We use the Bayesian information criterion as a figure of merit (see e.g. [22, 23]), and assume one model to be decisively favored over the other if FIG. 3: The best fit ΛCDM and Gaussian void models as p dashed and solid lines, with triangular and square data points, ∆ ln E ∆(ln max ln N) > 5, (7) respectively, for an example of 700 SNe simulated from a | | ≈ | L − 2 | ΛCDM model using the SALT light curve fitter. The shape of where E is the evidence for a model, max is the max- the redshift distribution used here is similar to that expected imum likelihood of a model, given a dataL set, p is the from the 2000 JDEM SNe, with an extra 300 at low z. We find number of parameters in the model, and N is the number that 700 SNe, with this redshift distribution, can decisively of data points. This criterion corresponds to one model recover the ΛCDM model 99% of the time, as it becomes evident that a void model cannot mimic the low redshift be- being 150 times more likely than the other. The mini- ∼ haviour of a ΛCDM cosmology. For illustrative purposes, we mal void model under consideration has 6 parameters: 2 have binned in groups of 60. to parametrize a Gaussian k, and 4 ‘nuisance’ parameters required to calibrate the SNe data. These are absolute Due to the different strategies and technologies, SNe magnitude, intrinsic error, and the colour and stretch pa- surveys typically target either low or high redshifts. This rameters used in light curve fitting, M0, σint,α,β . As- has lead to a dearth of SNe at 0.1 < z < 0.4 – exactly suming spatial flatness, ΛCDM requires{ 5 parameters:} 1 the location where there is the greatest∼ qualitative∼ differ- specifying the fraction of dark energy, and the same 4 nui- ence between the two models. Future SNe surveys, with sance parameters. In a more comprehensive study it may a redshift coverage in this region, will do better. As an be preferable to perform a full Bayesian evidence analy- example of the future constraints we can expect to gain, sis, with suitable priors [23]. In the interests of brevity, we consider the JDEM missions which expect to observe and to avoid a lengthy discussion of prior probabilities, 2000 high redshift SNe in the interval 0.1 < z < 1.7, we have refrained from this for now. ∼with an expected smooth distribution [27]. At∼ very∼ low In Figure 2 we show the SNLS data with the two best redshifts (0.03 < z < 0.1) a further 300 SNe can be fit models. Both have similar goodness of fit, but one can expected to be∼ observed∼ by other projects.∼ We consider discern a qualitative difference between them, which will data simulated from a ΛCDM model, using the SALT fit- 4 ter, and a Gaussian void model with the same parameter 300 SNe to the SNLS data, and letting them have a Gaus- values as before. In both cases we find that these 2300 sian redshift distribution with σz =0.1 and a mean that SNe are sufficient to decisively recover the correct under- can vary, we simulate the data from void and ΛCDM lying model. For the void model we find ∆ ln E = 89 12, models. We find that in a void Universe the optimal while for the ΛCDM model ∆ ln E = 46 10. These± are place to search for these extra 300 SNe is at z 0.3 (lim- considerably better than the decisive benchmark,± which iting ourselves to a mean redshift of less than∼ 1), where was satisfied by all 1000 of our simulations (except one in the average ∆ ln max 5. In a ΛCDM universe the SNe the ΛCDM case). We can therefore say with confidence would be betterL placed∼ a little lower, at z 0.1, and in ∼ that the upcoming SNe data will be able to distinguish this case ∆ ln max 2.5. ΛCDM from a void model, independent of which is re- We emphasiseL that∼ similar results to those presented sponsible for the apparent acceleration. above should be obtainable for any smooth void model. Given that 2300 SNe from a JDEM mission will do a For example, repeating our analysis for void (b), in Fig. superfluous job,∼ we will now estimate the minimum num- 1, gives almost identical results to void (a). The rea- ber of SNe required to meet the bound in Equation (7). son for this is that all smooth voids display the same Using the same redshift distribution as before [27], we qualitative behaviour of having a flat ∆dm(z) profile at now consider different numbers of SNe. In the case of low-z. It is this qualitative difference that allows the void the void cosmology we find that with 170 SNe 50% of models to be so easily distinguished from ΛCDM, and as our 1000 simulations recovered the void∼ model decisively, all smooth voids display this low-z behaviour, we expect while with 480 SNe 99% of the simulations could do so. the results we have presented to be broadly generalizable Similarly, in∼ the case of a ΛCDM cosmology we found to all voids. Of course, it may be possible to imagine that with 180 and 700 SNe we could decisively recover anomolous cases. This will be considered further in a the correct∼ model in∼ 50% and 99% of the simulations, re- more extended future publication. spectively. This is illustrated in Fig. 3. These projections Two very different paradigms have been invoked to ex- are much lower than the number of SNe expected to be plain the current observation of an apparently accelerat- observed by the next generation of SNe surveys, and we ing Universe, depending on whether we invoke or reject should therefore expect less ambitious projects to able to the Copernican Principle. We have shown that in the distinguish between the two models. In fact, this may be coming years it will be possible to experimentally distin- possible soon, with data from the Sloan Survey and third guish between these two scenarios, allowing us to exper- year SNLS expected to be released imminently. The re- imentally test the Copernican Principle [28, 29, 30], as sults obtained here depend on the details of the future well as determine the extent to which Dark Energy must surveys. Tailoring these to the specific regions where the be considered a neccessary ingredient in the Universe. two models differ most would undoubtedly give decisive Acknowledgements results sooner. Consider now the effect of varying the redshift distri- We are grateful to C. Clarkson, L. Miller, A. Slosar, bution, rather than the overall number. Adding a further and M. Sullivan for discussions, and the BIPAC for sup- port. TC acknowledges the support of Jesus College, and KL the Glasstone foundation and Christ Church.

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