Sudden Future Singularity models as an alternative to dark energy?

Abstract

1 INTRODUCTION

The discovery of the acceleration of the universe (Riess et al. 1998; Spergel et al. 2003) resulted in the breaking of the link between geometry and destiny of the universe. It was now the nature of the substance driving this acceleration that would determine the ultimate fate of the universe. Current observations are consistent with a cosmological constant as the origin of this acceleration, but they are not yet able to rule out other possibilities such as phantom energy which will drive the universe towards a ‘Big Rip’ singularity. This situation has encouraged the study of other exotic singularities. One such example is the Sudden Future Singularity (SFS, Type II according to the classification in Nojiri, Odintsov & Tsujikawa 2005), first proposed by Barrow (2004). Other exotic singularities discovered to date include the Finite Scale Factor (FSF) singularity (Type III, Nojiri et al. 2005), the Generalized Sudden Future Singularity (GSFS, Dabrowski, Denkiewicz & Hendry 2007), the Big Separation (Type IV, Dabrowski & Denkiewicz 2010) and the w-singularity (Dabrowski & Denkiewicz 2009). These singularities may serve as alternatives to dark energy (Dabrowski & Denkiewicz 2010). Furthermore, the Big Brake is a special kind of a singularity of the SFS type, which can arise in tachyonic models (Kamenshchik, Kiefer & Sandhofer 2007). In this paper we present the results of our investigation of one example class of models which accommodate an SFS, also proposed by Barrow, by confronting it with the current cosmological observations.

The present work is not the first time that the SFS models have been confronted with cosmological observations. Previously, Dabrowski et al. (2007) performed such a test using the luminosity distance redshift relation applied to SNe data. Their paper showed that these data were consistent with the SFS model over the redshift range probed by the SNe, and moreover could permit an SFS that would occur in a surprisingly short time: less than 10 Myr from now. In this paper we extend the analysis to confront this class of SFS models with some other available cosmological observations and constraints, from the cosmic microwave background (CMB) radiation, baryon acoustic oscillations (BAOs) and the age of the universe.

The structure of this paper is as follows. In Section 2 we introduce some relevant theory underlying SFS models. Section 3 gives an account of the cosmological probes we have used to constrain our SFS model, followed by an explanation of the methods we employed to perform these tests. In Section 4 we present the results of our investigations. Finally in Section 5 we give our conclusions.

2 SFS MODELS

In 2004 Barrow (2004) first published the results of his discovery of the existence of SFSs, which arise in the expanding phase of a standard Friedmann universe and violate the dominant energy condition only. This was an intriguing discovery since until then the theoretical search for expanding universes with possible violent ends had identified only Big Rip singularities which violate all the energy conditions (Barrow 2004). Consider the Friedmann equations:

1

2

where ρ is the energy density, p the pressure, a the scale factor, k the curvature index, c the speed of light, G the gravitational constant and an overdot represents derivative with respect to time. Barrow found that by keeping the scale factor and the Hubble expansion rate finite one will also necessarily maintain a finite density, but that the pressure can still diverge. The divergence of pressure is accompanied by the blow up of the acceleration, leading to a scalar polynomial curvature singularity (Hawking & Ellis 1973). Shortly after their discovery, Nojiri & Odintsov (2004a,b) showed that these singularities may be avoided (or moderated) when quantum effects are taken into account.

The finiteness of the energy density and the fact that p→∞ at an SFS means that the null (ρc²+p≥ 0), weak (ρc²≥ 0 and ρc²+p≥ 0), strong (ρc²+p≥ 0 and ρc²+ 3p≥ 0) energy conditions are satisfied but the dominant energy condition (ρc²≥ 0, −ρc²≤p≤ρc²) is violated.

We can see that the divergent behaviour of the pressure cannot be linked to the finite energy density or we will not have an SFS. Hence we need to release the assumption of an equation of state which imposes a link between these two quantities. It should be noted that an SFS can occur also in inhomogeneous and anisotropic models (Barrow 2004; Dabrowski 2005).

Similar to sudden singularities are the so called ‘quiescent’ singularities which occur in braneworld models (Shtanov & Sahni 2002). These are a type of sudden singularities whereby the pressure remains finite alongside the density and the Hubble parameter while higher derivatives of the scale factor diverge. Alam & Sahni (2006) confronted braneworld models with quiescent singularities with type Ia supernovae (SNe Ia) and BAO data and concluded that they would not fit observations. Furthermore the exact same SFS with the divergent pressure occurs in non-local cosmology as found by Koivisto (2008). The accelerating universe in this model will lead it towards an SFS rather than a de Sitter type epoch. Koivisto, however, shows that these singularities may be avoided by a slight modification of the non-local model.

Physically SFSs manifest themselves as momentarily infinite peaks of tidal forces, but geodesic completeness is satisfied in our SFS model. Hence SFS are regarded as weak singularities (Fernandez-Jambrina & Lazkoz 2004) which means infalling observers or detectors would not be destroyed by tidal forces (Dabrowski & Denkiewicz 2010). The universe will continue its expanding evolution beyond such weak singularities until for example the occurrence of a more serious singularity that is geodesically incomplete like the Big Rip which can end the universe (Dabrowski et al. 2007).

Barrow constructs an example SFS model where the scale factor takes the form:

3

where A > 0, B > 0, m > 0, C and n > 0 are free constants to be determined. By fixing the zero of time, i.e. setting a(0) = 0, and the time of the singularity, a(t_s) =a_s, the scale factor may be written in the equivalent form

4

Dabrowski et al. (2007) changed the original parametrization for the scale factor by using A=δa_s, to obtain

5

where a_s, n, m, δ and t_s are constants to be determined. This way they created a non-standardicity parameter, δ, which, as it tends to zero, recovers the standard Friedmann limit (i.e. a model without an SFS).

For a pressure derivative singularity to occur we need:

6

We note that, for r≥ 3, all energy conditions are fulfilled. These singularities are called GSFS (Dabrowski et al. 2007) and may occur in theories with higher order curvature corrections (Nojiri et al. 2005). For an SFS we need r = 2 which means that 1 < n < 2.

An important requirement is that the asymptotic behaviour of the scale factor close to the big bang singularity follows a simple power law a_BB=y^m which will simulate the behaviour of flat k = 0 barotropic fluid models with m = 2/3(w+ 1). This will ensure that all the standard observed characteristics of the early universe such as the CMB, density perturbations and big bang nucleosynthesis are preserved. All the energy conditions are satisfied if we require m to lie in the range

7

Furthermore, in accordance with (7), if for a standard dust-dominated universe we take m = 2/3 our SFS model will reduce to the Einstein-de-Sitter (EdS) universe at early times.

The other parameters of the model that we should consider are: a_s, t_s and δ. The parameter a_s, which sets the physical size of the universe at the time of the SFS, will cancel out in the equations of the standard cosmological probes we have used to test the model. Thus we do not need to consider this parameter further.

For constraining the time, t_s, when an SFS might occur, we can introduce the dimensionless parameter y₀=t₀/t_s, where t₀ is the current age of the universe. Since the singularity is assumed to be in the future, it follows that 0 < y₀ < 1.

The current acceleration obtained in the SFS model, as a result of the particular form adopted for the scale factor, leads to a divergence of the pressure at sometime in the future evolution of the model. Dabrowski & Denkiewicz (2010) refer to the cause of this late time acceleration in the SFS model as a ‘pressure-driven dark energy’.

With the parameters n, m and y₀ having definite values or ranges, it remained to identify a suitable range of investigation for the non-standardicity parameter δ. In Dabrowski et al. (2007) negative values of δ were associated with acceleration, but we revisited this question in order to check rigorously the range of values of δ which should be considered, taking into account all relevant physical constraints. In so doing we imposed the established observational facts that both the first and second derivatives of the scale factor are currently positive – i.e. we rejected any combination of SFS model parameters for which or ⁠. In addition we imposed the physical condition that positive and negative redshifts should correspond to past and future events, respectively. Finally we checked the sign of throughout the evolution of the SFS model and rejected parameter combinations that would predict contraction (i.e. ⁠) at any point in the interval 0 ≤z≤ 1, on the (conservative) grounds that the expansion of the universe is securely observed from e.g. the Hubble diagram of SNe Ia over this range of redshifts. We did not, however, make any further assumptions about the expansion history of the universe outside of this range; in particular parameter combinations that would predict e.g. future contraction of the universe were not excluded from our analysis.

Thus by fixing m = 2/3, as previously discussed, and varying n and y₀ over their permitted ranges, we sought to identify those values of δ which were consistent with the above physical constraints. It quickly became apparent that this task was not possible analytically. Therefore we carried out a numerical exploration of the three-dimensional parameter space (n, y₀, δ). This then told us that δ should not be positive.

3 COSMOLOGICAL CONSTRAINTS AND ANALYSIS METHODS

We carried out a comparison between our SFS model and current cosmological observations within a Bayesian framework, computing posterior distributions for the SFS model parameters inferred from each of the cosmological probes considered. From Bayes’ theorem we may write

8

where Θ denotes the parameter(s) of the SFS model and ‘data’ denotes generically the observed data for one of the cosmological probes under consideration. The term on the left-hand side of equation (8) is the posterior distribution for Θ, i.e. it describes our inference about the model parameters in the light of the cosmological data with which we are confronting the SFS model. The first term on the right-hand side is the likelihood function which gives the probability of obtaining the data that we actually observed, given a particular set of model parameters. The second term on the right-hand side is the prior probability distribution for the model parameters, based on e.g. theoretical considerations and/or previous observations. For each of our cosmological probes we took the likelihood function to be Gaussian in form, i.e.

9

where the constant of proportionality could be determined from the normalization conditions that the Likelihood function and posterior probability distribution should integrate to unity. In equation (9) the quantity χ² takes the general form

10

where x_obs is the observed value of the relevant cosmological probe, x_pred is the value predicted by our SFS model given the parameters Θ, σ is the uncertainty (due to measurement error and/or intrinsic variations) in the observed value and n is the number of data points associated with it. We now briefly describe each cosmological probe in more detail.

3.1 SNe Ia redshift–magnitude relation

We calculated the redshift–magnitude relation for our model to compare it with the observed relation as probed by SNe Ia as standard candles. Although as mentioned in the previous section an SFS could occur regardless of the curvature of the universe, to make our calculations simpler, and also to be consistent with the present observational data, we shall consider only a flat universe which bears an SFS. We used 557 SNe from the Union2 data set as compiled by Amanullah et al. (2010) which is the largest published SNe Ia sample to date. In our analysis we treated the Hubble constant (H₀) as a nuisance parameter, marginalizing over a range of values using as a prior distribution the most recent results from the Hubble Space Telescope (HST) Key Project (Riess et al. 2009), which assumes no underlying cosmology in measuring this parameter. We describe this marginalization process in more detail below. For our SNe Ia likelihood function we adopt the χ² quantity:

11

where μ denotes distance modulus, σ_phot is the measurement uncertainty tabulated for each SN and σ_int is the scatter intrinsic to every SN because they are not perfect standard candles. We take σ_int to be 0.15 as advocated by Kowalski et al. (2008).

Fig. 1 shows the distance modulus as a function of redshift for the SFS model and the concordance model for their respective best-fitting parameters and how they compare with the best SNe data set available. One can see that the SFS model fits the data very well and that it is almost distinguishable from the concordance model over the redshift range probed by the Union2 sample.

Figure 1

The predicted distance modulus plotted against redshift for the SFS model and Concordance Cosmology (CC) standard model. The SFS model prediction is shown as the solid cyan line and is for parameters n = 1.995, δ=−0.5, y₀ = 0.805. (These are the best-fitting SFS parameters for the Union2 SNe data set.) The concordance model prediction is shown as the dashed pink line and is for parameters Ω_m = 0.2725, Ω_Λ = 0.7275 (from the WMAP7 results; Komatsu et al. 2011). Both sets of calculations assume H₀ = 74.2 km s⁻¹ Mpc⁻¹ from the HST Key Project results (Riess et al. 2009). Also shown are the observed data points (with quoted 1σ errors) for the Union2 SNe data set. One sees that the SFS model fits the data very well and that the fit is almost indistinguishable from that for the concordance model. (A colour version of this figure is available in the online journal.)

Open in new tabDownload slide

3.2 CMB distance priors: R and l_a

In order to test a model against the CMB data one would ideally calculate the full angular power spectrum of the temperature anisotropies for the model, and compare it with the observed angular power spectrum. However, since this process is rather computationally intensive and complex, a simpler approach is instead to calculate the distance scales to which the power spectrum is very sensitive. The positions of the peaks and troughs of the CMB power spectrum, which can be measured precisely, provide a measure of the distance to the decoupling epoch. The distance ratios measured by the CMB are as follows.

1. The angular diametre distance to the last scattering surface (at redshift z_CMB, which we take to be 1089), D_A(z_CMB), divided by the sound horizon size at the decoupling epoch, r_s(z_CMB) which is quantified by the ‘acoustic scale’ and is defined by

   

   12

   where the factor (1 +z_CMB) accounts for the fact that D_A(z_CMB) is the proper angular diametre distance and we calculate r_s(z_CMB) following Wang & Mukherjee (2007).
2. The angular diametre distance to the last scattering surface divided by the Hubble horizon size at the decoupling epoch, which is called the ‘shift parameter’, R and is given by Komatsu et al. (2009):

   

   13

   where Ω_m is the matter density parameter. It was shown by Wang & Mukherjee (2007) that using these two parameters together is necessary in order to place tight CMB constraints on the parameters of the model of interest. They found that models with the same parameter R but different values for l_a, and vice versa, in general gave different CMB angular power spectra.

For the CMB constraints we therefore computed likelihoods with the following χ² expressions for the shift parameter and the acoustic scale, respectively:

14

15

The observed values for these quantities: R = 1.725 ± 0.018 and l_a = 302.09 ± 0.76, are obtained from the WMAP7 data and given by Komatsu et al. (2011). However, as pointed out by Elgaroy & Multamaki (2007) the shift parameter is not a directly measurable quantity and it is in fact derived from the CMB data assuming a specific cosmological model. Care therefore needs to be taken when using this quantity as a cosmological constraint.

The value for the shift parameter quoted above, as Komatsu et al. (2009) explains, has been derived assuming a standard FLRW universe with matter, radiation, curvature and dark energy components. The SFS model we are considering is also a standard Friedmann model but it assumes no explicit dark energy component; instead cosmic acceleration is driven by the divergence of pressure resulting from the particular form of scale factor adopted in the model. Matter is also permitted in the SFS and in fact, since we require our model to reduce to the EdS case at early times, we adopt the same matter content as that in the concordance model. Concerning the radiation and curvature components, we follow Komatsu et al. (2009) and ignore these in the shift parameter calculation.

Turning to the dark energy component, here we followed the approach of Elgaroy & Multamaki (2005) and expressed our SFS model in a form equivalent to an evolving dark energy model by computing its effective equation of state, w_eff(z). (See the appendix for a short derivation of the expression for w_eff.)

We then looked at the evolution of this effective w index to see how it compared with the observed behaviour. Fig. 2 shows the evolution of the effective equation of state over the redshift range 0 < z < 20, for two representative sets of SFS model parameters. In both cases we see the the same general features: w_eff≃−1 as z→ 0 and w_eff→ 0 for large z. These limiting behaviours are in good agreement with current observations. Note, however, that in plot (b) the particular SFS model parameters result in the divergence of w_eff at certain redshifts, which is caused by the denominator in the expression on the right-hand side of equation (A5) tending to zero at these redshifts. This behaviour is discussed in Shafieloo et al. (2006) where they make the case that in models where dark energy is not treated as an explicit fluid or a field, the equation of state cannot be used as a fundamental quantity and indeed an effective equation of state may display unusual properties like singularities. Very similar divergent behaviour was found e.g. by Sola & Stefancic (2005) in computing an effective equation of state for their evolving Λ model; indeed those authors refer explicitly to each divergent feature as a ‘fake singularity, which is nothing but an artefact of the EOS parametrization’. In our case too the divergence of w_eff is not seen as an indication of a fundamental physical problem with the model. Nevertheless the general similarity of the limiting behaviour of w_eff to that of the concordance model gives us confidence, following Elgaroy & Multamaki (2007), that it remains appropriate to use the ‘observed’ value of the shift parameter when investigating our SFS model.

Figure 2

The evolution of w_eff as a function of redshift in the SFS model, for two representative sets of model parameters. Both plots show generic behaviour as z→ 0 and z→∞ which is in good agreement with current observations, although plot (b) also shows ‘fake’ singularities due to the parametrization of w_eff.

Open in new tabDownload slide

3.3 BAO distance parameter, A

BAOs which originate from the excitation of sound waves in the early universe photon-baryon plasma through cosmological perturbations are useful distance indicators at the current epoch. We can use this to constrain very well the following quantity, termed as the ‘distance parameter’, using the current data:

16

The function E(z) is defined as E(z) =H(z)/H₀, where H(z) is the Hubble expansion rate. z_BAO is the effective redshift of the galaxy sample used to measure the distance parameter and it is 0.35 for the SDSS galaxy sample. For the BAO distance parameter constraint we adopt:

17

Here the observed value for the distance parameter has been taken from the latest SDSS results which is: A = 0.469(n/0.98)^−0.35± 0.017 (Eisenstein et al. 2005). We used the value of n = 0.963 for the spectral index from WMAP7 results (Komatsu et al. 2011).

We find that the BAO distance parameter is not immune to the model dependency issue in its derivation either. This issue is considered in detail in e.g. Carneiro et al. (2008) who, following Eisenstein & Hu (1998), identify two (implicitly assumed) conditions which should be valid in order that the BAO distance parameter is applicable. For the model in question firstly the evolution of matter density perturbations during the matter dominated era must be similar to the CC case, and secondly the comoving distance to the horizon at the epoch of matter-radiation equality should scale inversely with Ω_mH²₀. While these conditions are not met for the Carneiro et al. model of a time varying cosmological constant, they are met in our case.

3.4 Age of the universe, t₀

Using the standard Friedmann equation for calculating the age of the universe we have

18

where D_H is the Hubble distance and E(z) is defined as before. In our likelihood function for this constraint we used the χ² expression:

19

where t^pred₀ is the age of the universe predicted in our SFS model, as computed from equation (18). For t^obs₀, the observed age of the universe, we followed e.g. Balbi, Bruni & Quercellini (2007) and used the best current estimate derived from various astrophysical probes, including globular cluster ages, that have been determined without assuming a particular cosmological model. We adopt t₀ = 12.6^+3.4_−2.2 Gyr with an asymmetric error distribution corresponding to 95 per cent upper and lower confidence limits, as reported in Krauss & Chaboyer (2003).

3.5 Hubble constant, H₀

The current value of the H₀ is the final constraint which we employed in our analysis. Although the H₀ already features indirectly in the SNe Ia and age constraints, since it enters into our calculations of the predicted values for those two probes, we considered it useful also to compare directly the observed value of H₀ with its predicted value calculated for our SFS model. The latter is simply given by

20

We thus computed a likelihood function for this constraint using the χ² quantity

21

where H^obs₀ = 74.2 ± 3.6 ⁠, i.e. the (cosmology independent) HST Key project value discussed earlier.

4 RESULTS

In this section we present posterior distributions for our SFS model parameters inferred from each of the cosmological probes described in Section 3. For ease of presentation we have computed a series of ‘slices’ through the two-dimensional conditional distribution of the parameters n and y₀ at given values of the non-standardicity parameter δ. We adopted uniform priors for n and y₀ over the intervals 1 < n < 2 and 0 < y₀ < 1, respectively, on the theoretical grounds discussed in Section 2. Thus, for these uniform priors, the mode of the posterior distribution function, for each ‘slice’ in δ was coincident with the maximum of the conditional likelihood function.

For the case of the SNe Ia data, as noted in Section 3 we computed the posterior distribution after first marginalizing over the H₀. It follows straightforwardly from Bayes’ theorem that

22

where p(H₀) represents our prior information on the H₀. Thus for the SNe Ia case we computed p(n, y₀|δ) following equation (22), adopting a Gaussian prior for H₀ with a mean value of 74.2 and a standard deviation of 3.6, in accordance with the results of the HST Key Project.

We computed posterior distributions adopting both a regular grid of (n, y₀) values over the two-dimensional parameter space and a simple MCMC-based approach using the Metropolis algorithm. Both methods gave very similar results, although the latter approach is more practical when exploring more general parameter spaces with larger numbers of parameters. We will consider such cases further in a subsequent paper.

Figs 3–5 present contour plots showing the Bayesian credible regions (evaluated at the 68, 95 and 99 per cent level) for n and y₀, computed for each of our cosmological probes at a series of fixed values of δ. In each figure we show separately contours for each of the six cosmological probes we have considered, labelled (a)–(f), respectively. These probes are: (a) the SNe Ia redshift-magnitude relation; (b) the CMB shift parameter, R; (c) the CMB acoustic scale, l_a; (d) the BAO distance parameter, A; (e) the present-day value of the H₀; (f) the present age of the universe, t₀. Fig. 3 shows results for δ=−0.7. For larger (i.e. less negative) values of δ the posterior distributions for most of the probes were qualitatively similar; significantly, however, the predicted value of the shift parameter was such that no credible region for the SFS model parameters was found at the 99 per cent level. We return to this point below.

Figure 3

The contours show 68, 95 and 99 per cent credible regions in the (n, y₀) parameter space for a fixed value of δ=−0.7. The plots labelled (a), (b), (c), (d), (e) and (f) correspond to the contours calculated using the probes, SNe Ia redshift-magnitude relation, CMB shift parameter, R, CMB acoustic scale, l_a, BAO distance parameter, A, the H₀ and the age of the universe, respectively.

Open in new tabDownload slide

Figure 4

The contours show 68, 95 and 99 per cent credible regions in the (n, y₀) parameter space for a fixed value of δ=−1. The labels are as described in Fig. 3.

Open in new tabDownload slide

Figure 5

The contours show 68, 95 and 99 per cent credible regions in the (n, y₀) parameter space for a fixed value of δ=−1.5. The labels are as described in Fig. 3.

Open in new tabDownload slide

One can clearly see from Fig. 3 that there is no part of the (n, y₀) plane where the credible regions for the six cosmological probes overlap, for this value of δ. There is substantial overlap between the SNe Ia and BAO contours; this is not surprising given that both probes are sensitive to the E(z) function over a similar range of redshifts.

The age of the universe constraint (f) is rather weak, reflecting the large uncertainty on the observed value, and only a small region of the (n, y₀) plane, on the upper left of Fig. 3(f), is excluded at the 95 per cent level; these parameter values are already strongly excluded by all other probes.

In Fig. 3(b) we have magnified the shift parameter contours for greater clarity. Note, however, that the minimum χ² value for the shift parameter in this plot is already 21.35 (corresponding to a predicted value of R = 1.641) which is unacceptably large – i.e. the likelihood function (and hence the posterior probability) for the SFS model parameters is everywhere vanishingly small for the shift parameter at this value of δ. Similar behaviour was observed for larger values of δ; indeed the minimum χ² value becomes progressively higher as δ increases. Hence we do not show contour plots for δ > −0.7.

It is interesting to note from panels (b) and (c) in Fig. 3 that the two CMB constraints R and l_a produce credible regions which do not in fact show any overlap at the 99 per cent level. This illustrates the importance of the point made by Wang and Mukherjee, as noted earlier, that one should use both CMB probes in order to better constrain (or indeed, as is the case here, to reject) a given model, since these probes are sensitive to the model parameters in different ways. Moreover we also see that the credible regions for R and l_a do not share a common overlap with any of the other four probes either. Hence a fit to the SFS model parameters is strongly excluded for this value of δ.

Figs 4 and 5 present credible regions for two further values of δ, in order to illustrate the pattern of behaviour exhibited by the various probes as δ decreases. First Fig. 4 shows results for δ=−1. We can see that, as before, there appears to be overlap between the credible regions for SNe, BAO distance parameter, H₀ and age of the universe; indeed the contours for these probes have shifted only slightly from their position in Fig. 3. However all four show no overlap at the 99 per cent level with either of the CMB probes, which in turn continue to show no overlap with each other.

Finally, Fig. 5 shows the credible regions for δ=−1.5. Here we can see that the contours follow more or less the same shapes as in the previous figures. There is still no overlap between R and l_a, and neither CMB probe overlaps with any of the other probes, again at the 99 per cent level. Note, moreover, that while the general appearance of the credible regions is similar to Fig. 4, the contours for the acoustic scale, l_a, in Fig. 5(c) have shifted slightly downwards and appear to be approaching the lower limit of y₀. As δ is decreased further this trend continues and hence no fit is obtained. To emphasize that indeed the contours display no overlap with one another, we show in Fig. 6 the most important contours of SNe Ia, CMB shift parameter, R, CMB acoustic scale, l_a and BAO distance parameter superimposed for values of δ previously considered in Figs 3–5. One can see clearly in these figures that while the SNe Ia data are consistent with an SFS occurring in the very near future, as shown by Dabrowski et al. (2007), the same SFS parameters would not give a fit to the CMB data.

Figure 6

The panels show a superposition of the credible regions in the (n, y₀) parameter space derived from the SNe Ia, CMB shift parameter, R, CMB acoustic scale, l_a and BAO distance parameter, A data, for the values of δ=−0.7 (a), δ=−1 (b) and δ=−1.5 (c), as previously illustrated in Figs 3–5, respectively. Note that in each panel there is no part of the parameter space where all credible regions overlap. (A colour version of this figure is available in the online journal.)

Open in new tabDownload slide

Note that for the purpose of straightforward illustration the contour plots presented in Figs 3–5 have been calculated first without the imposition of the physical constraints discussed in Section 2. A typical example of the application of these conditions, and their impact on the credible regions, is shown in Fig. 7 for the acoustic scale, l_a. One can see in Fig. 7(b) how the contours are abruptly cut-off, in this case by the inclusion of the condition that the universe currently be accelerating. In other words, in the region enclosed by the contours in the right-hand part of Fig. 7(a), e.g. for y₀ < 0.6 and n > 1.5, our SFS model predicts a value of l_a which is in satisfactory agreement with observations, but for a universe which is not currently accelerating. The other cosmological probes such as the age of the universe and the BAO distance parameter are similarly affected, although much less severely, for certain values of δ. However, since it is already the case that we find no significant overlap between the credible regions over the entire parameter space without applying these additional physical constraints, we will not present any further plots that do include them.

Figure 7

The impact of the imposition of the physical conditions discussed in Section 2 is shown here for the l_a contours. The plot labelled (a) shows these contours not considering the physical conditions while in plot (b) these conditions are included.

Open in new tabDownload slide

5 SUMMARY AND CONCLUSIONS

In this paper we have investigated one class of SFS models proposed by Barrow, by confronting it with the currently available observational data. After introducing the theory behind the model and explaining its characteristics in Section 1, we reported on the cosmological probes we used in testing our model in Section 2, taking care throughout to consider thoroughly the applicability of each probe to the SFS model under study. We then presented the results of our investigations in Section 4. Specifically we found that the SFS model was not compatible with all of the current cosmological observations considered: while good agreement could be found with e.g. the Union2 SNe data and BAO distance parameter at low redshift, the same model parameters predicted an acoustic scale and a shift parameter for the CMB which were in strong disagreement with WMAP7 observations.

We have presented our results as a series of conditional ‘slices’ through the (n, y₀) parameter space at fixed values of δ. We note that the rejection of the SFS model does not depend on our choice of two-dimensional parameter space on which to conduct our analysis. The choice of the (n, y₀) space was convenient because of the ranges of these two parameters, but the absence of any region which gave a fit to the model is a robust result that can be extended to the full three-dimensional space of (n, y₀, δ). In other words, by showing that there is no fit to the data in conditional two-dimensional spaces we have therefore shown that there is no fit to the data in the full three-dimensional parameter space.

In conclusion we note that our results are for a fixed value of m = 2/3, which ensures that our SFS model reduces to the EdS universe at early times. In a forthcoming paper (Dabrowski et al. in preparation) we will relax this assumption and consider SFS models in which the parameter m is allowed to vary, by confronting these models with the same cosmological observations which we have considered here.

HG would like to thank Dr Arman Shafieloo for his useful comments and suggestions.

REFERENCES

Appendix

APPENDIX A: DERIVATION OF w_eff

In this section we present a short derivation of the expression for w_eff discussed in Section 3.2. A dark energy component with equation of state parameter w(a), will correspond to a density that varies with the scale factor according to

(A1)

The standard Friedmann equation for a flat model will therefore take the form

(A2)

Rewriting the above using a = (1 +z)⁻¹ we have

(A3)

which means:

(A4)

where E(z) =H(z)/H₀. Now, differentiating both sides gives, where ′≡ d/d z:

(A5)