---
abstract: |
    Cosmic acceleration is investigated through a kink-like expression for the deceleration parameter ($q$). The new parametrization depends on the initial ($q_i$) and final ($q_f$) values of $q$, on the redshift of the transition from deceleration to acceleration ($z_{t}$) and the width of such transition ($\tau $). We show that although supernovae (SN) observations (*Gold182* and SNLS data samples) indicate, at high confidence, that a transition occurred in the past ($z_{t}>0$) they do not, by themselves, impose strong constraints on the maximum value of $z_{t}$. However, when we combine SN with the measurements of the ratio between the comoving distance to the last scattering surface and the SDSS+2dfGRS BAO distance scale ($S_{k}/D_{v}$) we obtain, at $95.4\%$ confidence level, $z_{t}=0.84\pm _{0.17}^{0.13}$ and $\tau =0.51\pm
    _{0.17}^{0.23}$ for ($S_{k}/D_{v}$+*Gold182*), and $%
    z_{t}=0.88\pm _{0.10}^{0.12}$ and $\tau =0.35\pm _{0.10}^{0.12}$ for ($%
    S_{k}/D_{v}$ + SNLS), assuming $q_i=0.5$ and $q_f=-1$. We also analyze the general case, $q_f\in(-\infty,0)$ finding the constraints that the combined tests ($%
    S_{k}/D_{v}$ + SNLS) impose on the present value of the deceleration parameter ($q_0$).
address: 'Universidade Federal do Rio de Janeiro, Instituto de Física, CEP 21941-972, Rio de Janeiro, RJ, Brazil'
author:
- 'Émille E. O. Ishida'
- 'Ribamar R. R. Reis'
- 'Alan V. Toribio'
- Ioav Waga
title: 'When did cosmic acceleration start? How fast was the transition?'
---

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Introduction
============

Since the discovery of the accelerated expansion of the universe in 1998 [@riess98; @perlm99], considerable effort in cosmology has been devoted to determine the source of this acceleration. The two most common possibilities discussed in the literature are: the existence of an exotic component with sufficiently negative pressure (dark energy) and proper modifications of general relativity at cosmological scales (for recent reviews see [@review]).

One way of making progress in determining the cosmic expansion history is through a model by model analysis. Another is to carry out a phenomenological analysis with the use of different parameterizations of the dark energy equation of state [@eosparam], the Hubble parameter [@hubble] or the dark energy density [@wang01]. This procedure may provide interesting pieces of information, but in general a parametrization assumes the existence of dark matter and dark energy as different substances (barring a few exceptions no interaction in the dark sector is considered) and general relativity is in most cases assumed. In this framework, an important question regards the number of parameters necessary to get reliable conclusions. If too many are used, the allowed region in the parameter space could be so large that it would not be possible to get firm conclusions [@linder05]. Otherwise, if not enough parameters are used, the obtained results may be strongly dependent on the particular parametrization choice and misleading conclusions could be reached [@bassett04]. The strategy we follow here is to use a large (four) number of parameters in order to be quite general, but, based on physical arguments, fix two of them from the start. We then relax the condition on one of the fixed parameters and obtain the confidence surface on the other three.

In this work we are mainly interested in the following questions: what is the redshift of the transition from decelerated to accelerated expansion? How fast was it? We investigate these by introducing a new parametrization for the deceleration parameter ($q$) that depends on four parameters: the initial ($q_i$) and final ($q_f$) values of $q$, the redshift of the transition from deceleration to acceleration ($z_{t}$) and a quantity related to the width in redshift of such transition ($\tau $). With this formulation we aim to answer the above questions with the minimum amount of assumptions about the dark sector and the fundamental gravitation theory.

This paper is organized as follows: in Section 2, we present the new $q$ parametrization and discuss some of its properties. In Section 3 the outcomes of the confrontation of this parametrization with two supernovae samples, the new *Gold182* SNe Ia from [@gold] and the first year data set of the Supernova Legacy Survey (SNLS) [@snls], are obtained (first assuming $q_f=-1$ and $q_i=1/2$). We show that current supernovae observations alone are not able to satisfactorily constrain the transition redshift. To break the SN Ia degeneracy, we combine this observable with the ratio of the comoving distance to the last scaterring surface ($S_{k}(z_{ls}=1098)$) to the baryon acoustic oscillations (BAO) distance scale ($D_{v}(z)$) at $ z_{BAO}=0.2$ and $z_{BAO}=0.35$ as estimated in [@percival07]. We show that the $\Lambda$CDM model is within the region allowed by SNLS+$S_k/D_v$ results but is excluded for *Gold182*+$S_k/D_v$ data, at $95\%$ confidence level. We then discuss the broader case with arbitrary $q_f$ exhibiting the $95\%$ confidence surface in the parameter space $(z_t,\tau,q_0)$ ($q_0$ is the present value of $q$), obtained using the SNLS+ $S_{k}/D_{v}$ data. Our conclusions are presented in Section 4.

The Model
=========

At large scales, it is a good approximation to consider a spatially homogeneous and isotropic universe. With this assumption we are lead to the Friedman-Robertson-Walker metric: $$ds^{2}=dt^{2}-a(t)^2\left[ \frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^2
\right] ,$$where $a(t)$ is the scale factor and $k=-1,0,+1$ characterizes the curvature of the spatial sections of space-time. From now on we will assume a flat universe ($k=0$), which is in agreement with CMB results [@spergel].

In terms of the Hubble parameter ($H\equiv \frac{\dot{a}}{a}$), the deceleration parameter can be written as: $$q=-\frac{\ddot{a}}{aH^{2}}=\frac{d}{dt}\left( \frac{1}{H}\right) -1.$$Therefore, $$H=H_{0}\exp {\left[ \int_{0}^{z}(q(\tilde{z})+1)d\ln
{(1+\tilde{z})}\right] }. \label{H}$$

In this work, we propose the following phenomenological functional dependence with redshift for the deceleration parameter: $$q(z)\equiv q_{f}+\frac{(q_{i}-q_{f})}{1-\frac{q_{i}}{q_{f}}\left( \frac{%
1+z_{t}}{1+z}\right) ^{1/\tau }},  \label{novoq}$$where $q_{i}>0$ (deceleration) and $q_{f}<0$ (acceleration) are the initial $%
(z\gg z_{t})$ and final $(z=-1)$ values of the deceleration parameter, respectively. The parameter $z_{t}$ denotes the redshift of the transition ($%
q(z_{t})=0$) and $\tau >0$ is associated with the width of the transition. It is related to the derivative of $q$ with respect to the redshift at $z=z_{t}$. More precisely, $$\tau ^{-1}=\left(\frac{1}{q_{i}}-\frac{1}{q_{f}}\right)\left[\frac{dq(z)}{d\ln (1+z)}%
\right]_{z=z_{t}}.  \label{tau}$$The influence of parameters $z_t$ and $\tau$ are demonstrated in Fig. (\[fig3\]).

Expression (\[novoq\]) is similar in spirit to the one suggested in [bassett]{} (see also [@linder05; @bassett04; @kink]), but here we parametrize $q(z)$ instead of $w(z)$. One of the advantages of using the above kink-like parametrization for the deceleration parameter is that $z_{t}$ has a very clear physical meaning. Different physical aspects and parameterizations of $q(z)$ were also investigated in [@qparam].

With the above definition, equation (\[H\]) is now integrated to give, $$\begin{aligned}
\left( \frac{H(z)}{H_{0}}\right) ^{2}&=&\left( 1+z\right)
^{2(1+q_{i})} \nonumber \\
& & \times \left( \frac{q_{i}\left( \frac{1+z_{t}}{1+z}\right)
^{1/\tau }-q_{f}}{q_{i}\left( 1+z_{t}\right) ^{1/\tau
}-q_{f}}\right) ^{2\tau (q_{i}-q_{f})}. \label{hq0}\end{aligned}$$ We now define an effective matter density parameter ($\Omega
_{m\infty }$) as $$\Omega _{m\infty } \equiv \lim_{z\rightarrow \infty }\left( \frac{H(z)}{%
H_{0}}\right) ^{2}\left( 1+z\right) ^{-2(1+q_i)},$$where the limit should be understood as $z>>z_t$.

In most (and simplest) scenarios, in order to form large scale structures, the universe passes trough a kind of matter dominated phase such that, at early times (but after radiation domination), $H^{2}\propto (1+z)^{3}$, which implies $q=1/2$. In this work we fix $q_{i}=1/2$ reducing to three the number of free parameters. In principle, with this assumption we are losing generality but the question is: how much $q_{i}$ can deviate from $1/2$ during large scale structure formation? In the general relativity framework, in some models with a constant coupling ($\delta$) between dark matter and dark energy this condition ($q_{i}= 1/2$) is not satisfied. In this case we have $H^{2}\propto (1+z)^{(3+\delta)}$ during matter domination. But, what are the allowed values for $\delta$? In [@guo07] it has been shown that background cosmological tests, impose |$\delta| < 0.1$. Taking into account matter perturbations stronger constraints on the coupling can be obtained [@fabris]. Another possibility is to consider models in which matter has pressure, such that when it dominates $q_{i}\neq 1/2$. However, if matter perturbations are adiabatic, due to a finite speed of sound, the mass power spectrum will present instabilities ruling out these models unless $p=0$ ($q=1/2$) or very close to it. In principle, it is possible to circumvent this kind of problem by assuming entropy perturbations such that $\delta p=0$ [@reis]. However, in this case the models may have problems with lensing skewness as pointed out in [@reis2]. Although there are some indications that by fixing $q_{i}=1/2$ we are not losing much in our description of the majority of the viable models, relaxing this condition requires further investigation, and we leave it for future work.

In the specific case $q_{i}=1/2$ we have , $$\Omega _{m\infty }=\left( 1-\frac{1}{2q_{f}}\left( 1+z_{t}\right) ^{1/\tau }\right) ^{-\tau
(1-2q_{f})}.  \label{omeff}$$With the above definition, we can eliminate $z_{t}$ from equation (\[hq0\]) and rewrite it as $$\begin{aligned}
&&\left( \frac{H(z)}{H_{0}}\right) ^{2}=\left( 1+z\right) ^{3} \notag \\
 \times  &&\left( \Omega _{m\infty }^{\frac{1}{\tau (1-2q_{f})}}+(1-\Omega
_{m\infty }^{\frac{1}{\tau (1-2q_{f})}})(1+z)^{-\frac{1}{\tau
}}\right) ^{\tau (1-2q_{f})}.  \label{hq1b}\end{aligned}$$The above expression for $H(z)$ in terms of $\Omega _{m\infty }$ is very useful to make connections with models already discussed in the literature. For instance, it is simple to verify from (\[hq1b\]) that the parametrization (\[novoq\]), in the special case $q_{i}=1/2$, is related to the Modified Polytropic Cardassian (MPC) model [@gondolo]. This model depends on three parameters: $m$ (denoted by $q$ in [@gondolo]), $n$ and $\Omega _{m0}$. If we identify, $%
\Omega _{m0}=\Omega _{m\infty }$, $m=1/(\tau (1-2q_{f}))$ and $n=2/3(1+q_{f})
$, it follows that the two models have the same kinematics. Note that, since $%
q_{f}<0$, the condition $n<2/3$ follows naturally. We remark that in the MPC model, $%
\Omega _{m0}$ is the present value of the matter density parameter, while $\Omega _{m\infty }$ is defined at high redshift. These two quantities do not necessarily have the same value in the general case if, for instance, dark matter and dark energy are coupled [@coup]. As an example, consider models with a variable coupling between dark matter and dark energy (assumed to have constant equation of state $w_x$), and such that $\rho_X/\rho_m=\rho_{X0}/\rho_{m0} a^{\xi}$ [@dalal]. These models can be described by Eqn. (\[hq1b\]) if we identify $\Omega_{m0}=\Omega _{m\infty }^{1/\tau(1-2q_f)}$, $\xi=1/\tau$ and $w_X=-(1-2q_f)/3$. As remarked before, in our formulation it is not necessary to make strong assumptions about the dark sector or gravity theory. In MPC model the universe components are specified to be matter and radiation; there is no dark energy. The parametrization (\[novoq\]) includes the MPC model (and the coupling models above) as special cases.

Neglecting baryons, the quartessence Chaplygin model $(p=-M^{4(\alpha +1)}/\rho ^{\alpha
})$ [@quartessence], is obtained if we assume $q_{i}=1/2$, $q_{f}=-1$, identify $1/\tau =3(1+\alpha )$ and $\Omega _{m\infty
}=(1-w_{0})^{1/(1+\alpha )}$, where $w_{0}=-M^{4}/\rho _{0}^{\alpha
+1}$ is the present value of the equation of state parameter.

The conventional dark energy model with constant equation of state ($w_{X}$) is obtained if we identify $\Omega _{m\infty }=\Omega
_{m0}$ and impose the condition $-3w_{X}=1/\tau =(1-2q_{f})$ in Eqn. (\[hq1b\]). In particular, if $q_{f}=-1$ and $\tau =1/3$, $\Lambda$CDM is recovered. For this model the transition redshift is equal to $(2(1-\Omega _{m0})/$ $\Omega _{m0})^{1/3}-1$. Identifying $\Lambda$CDM in the parameter space is very convenient; it fits current data quite well and we should expect the true cosmology not to be far from this limit. We remark that, in the framework of general relativity with non-interacting dark matter and dark energy, if $\tau <1/3$ and $q_{f}=-1$, the dark energy component will present a transient phantom ($%
w<-1$) behavior, that could either have started in the past or in the future ($z<0$). Models with $\tau >1/3$ are always non-phantom.

It is curious that if we apply the definition of $\tau $, given by Eqn. ([tau]{}) (assuming $q_{i}=1/2$ and $q_{f}=-1$), to the flat DGP brane-world model [@dgp] we obtain $\tau =1/2$, independent of $\Omega _{m0}$. Therefore q-models with $z_{t}=(2(1-\Omega
_{m0})^{2}/$ $\Omega _{m0})^{1/3}-1$ (the DGP redshift transition) and $\tau \approx 1/2$ are expected to be a good approximation for flat DGP models.

Observational Constraints
=========================

One of the main questions today in cosmology is to know if cosmic acceleration is generated by a cosmological constant or not. The data seem to indicate that models close to $\Lambda$CDM are favored. In our analysis we first consider the special case of models that have a final de Sitter phase ($q_{f}=-1$). In this case, the flat $\Lambda$CDM model is more easily identified in the parameter space allowing a simple test of the $\Lambda$CDM paradigm. The more general case, with arbitrary $q_{f}$, will also be briefly considered.

Assuming $q_f=-1$ we now derive constraints on the parameters $\tau
$ and $z_{t}$ by combining supernovae measurements with the ratio of the comoving distance to the last scaterring surface, $S_{k}(z_{ls}=1098)$, to the BAO distance scale, $D_{v}(z)$, at $z_{BAO}=0.2$ and $z_{BAO}=0.35$, as estimated in [@percival07]. In fact, the ratio $S_{k}/D_{v}$ times $z_{BAO}$ is equal to the ratio of the CMB shift parameter ($\mathcal{R}$) [@shift] at $z_{ls}$ to the BAO parameter $\mathcal{A}(z_{BAO})$ [@eisenstein]. This observable is appropriate for our purpose for two reasons. First, it does not explicitly depend on the exotic dark constituents of the universe and neither on the gravity theory. It is essentially controlled by the function $H(z)/H_{0}$. Second, complementarity with the SN constraints is generated because the $S_{k}/D_{v}$ ratio and SN are sensitive to distances to objects (events) in different redshift range; with supernovae we are measuring distances up to $z\sim 1-2$, while $S_{k}$ depends on the comoving distance to $z\sim 1100$.

In the SN Ia analysis we considered both, the *Gold182* [@gold] and the SNLS [@snls] samples. To determine the likelihood of the parameters we follow the same procedure described in these two references. In our computations, when marginalizing over the Hubble parameter, we use a Gaussian prior such that $h=0.72\pm 0.08$ [@hst]. In Fig.\[fig1\] (left-panel), assuming $q_f=-1$, we display constant confidence contours (68% and 95%) in the ($\arctan z_{t},\tau)$ plan allowed by SN experiments. Notice that, for the two SN data sets $z_{t}<0$ is not allowed at a high confidence level, indicating that a transition occurred in the past. We remark that this is expected in $\Lambda$CDM models (or other models) that have a non-null transition time, but our results indicate that this is true even if the transition is instantaneous ($\tau =0$). This conclusion also applies if $q_f\neq-1$. Furthermore, it is also clear in Fig.\[fig1\] (left-panel) that current SN observations cannot impose strong constraints on the maximum value allowed for $z_{t}$. Since SN observations prove the universe only up to redshift $z\sim
1-2$, in a model in which the transition is slow ($\tau \gtrsim 1$), even if $z_{t}$ is high, the distance to an object, let say, at $z\lesssim 1$, can be similar to the distance to the same object in another model in which $z_{t}\lesssim 1$ with a faster transition (smaller $\tau $). This explains the shape of the SN contours. By comparing the confidence contours for the two data sets, we observe that those from *Gold182* are shifted to lower $z_t$ with respect to those from SNLS. We remark that even in the region of more interest ($z_{t}\lesssim 1$), the difference between the outcome of the two SN Ia data sets, although not so severe, exists and is important. Similar results were obtained in [@nesseris07], which can be related to possible inhomogeneities present in the *Gold182* sample and should be further investigated.

To obtain the constraints on the parameters from the $S_{k}/D_{v}$ test, we use a $\chi ^{2}$ statistics taking into account the correlation matrix and the ratio $r_{s}/S_{k}$ given in [@percival07]. Since we are assuming flat space we have, $D_{v}(z_{BAO})=[z_{BAO}H^{-1}( \int_{0}^{z_{BAO}}
d\tilde{z}{H)^{2}}] ^{1/3}$ and $
S_{k}=\int_{0}^{1098}d\tilde{z}H_{e}^{-1}$. The $H_e$ term in the definition of $S_k$ incorporates the necessity of taking into account the contribution of radiation at very early times, which is not included in our parametrization Eqn. (\[novoq\]). One may argue that by introducing radiation we are losing generality. However, any late time modification of the standard cosmological model should satisfy Big-Bang nucleosynthesis (BBN) constraints at very early time. We know that radiation exists and what should be the dependence of the Hubble parameter with redshift at very early times (when radiation dominates) in order not to spoil BBN’s success. During this phase $H^2\propto (1+z)^{4}$ and $q=1$. Therefore, to take radiation into account, we add the term $\Omega
_{r0}(1+z)^{4}$ to the right hand side of (\[hq0\]) when applying it to calculate $S_k$. If we do not consider it, we would have a $\sim 18\%$ error in estimating ${S}_{k}$.

We marginalize the likelihood over $h$ with the same Gaussian prior used in the supernovae analysis. In fact, $S_{k}$ is almost independent of $h$; the dependence entering only through the radiation term. In Fig.\[fig1\] (right-panel) we show constant confidence contours (68% and 95%) in the ($z_{t},\tau $) plan allowed by the $S_{k}/D_{v}$ test. It is worth to be mentioned that the shape of the $95\%$ contour is similar to contours of constant $\Omega _{m\infty }$ and we can think that this test essentially constrains this quantity. The same kind of behavior also appears in flat, constant $w$ models [@percival07]. Furthermore, it is clear from the figure the complementarity between the SN and this test.

To get the combined (SN+$S_{k}/D_{v}$) results we multiply the marginalized likelihood functions. In Fig.\[fig2\] (left-panel) we show the results ($68\%$ and $95\%$ c.l.) of the $S_{k}/D_{v}$ test with the *Gold182* (green dashed contours) and with the SNLS (blue dot-dashed contours) data set. The red horizontal line in the figure represents the $\Lambda$CDM limit ($\tau =1/3$). It shows that this model is in good agreement with the $S_{k}/D_{v}$ + SNLS data. After marginalizing over the extra parameter [@betocs] we find for $S_{k}/D_{v}$+*Gold182* (at $95.4\%$ confidence level), $z_{t}=0.84\pm _{0.17}^{0.13}$ and $\tau =0.51\pm
_{0.17}^{0.23}$, while for $S_{k}/D_{v}$+SNLS we have, $z_{t}=0.88\pm _{0.10}^{0.12}$ and $\tau =0.35\pm _{0.10}^{0.12}$. For a model with $q_f=-1$, $\tau = 0.35$ and $z_t=0.88$, we obtain from Eqn. (\[omeff\]) that $\Omega _{m\infty }=0.23$. It is also simple to show that the age of the universe in this particular model (assuming $h=0.72$) is $14.0$ $Gyr$ and that cosmic acceleration started $7.2$ $Gyr$ ago. Notice that the $\Lambda$CDM cosmology ($\tau =1/3$) is in good agreement with $S_{k}/D_{v}$ + SNLS, but excluded at $95.4\%$ confidence level by the $S_{k}/D_{v}$+*Gold182* data. This discrepancy reveals tension between the two SN data samples and reinforces the necessity of better SN data to clarify the issue. We also display in the same figure (solid contour) what should be expected from future surveys when combining SN Ia+$S_{k}/D_{v}$ measurements. In our Monte Carlo simulations we used as fiducial model a flat $\Lambda$CDM model with $\Omega _{m0}=0.23$ ($\tau =1/3,z_{t}\simeq 0.88$). For SN Ia we considered a SNAP-like survey assuming that the intercept is known. For the $S_{k}/D_{v}$ test we used a conservative (but somewhat arbitrary) assumption that the uncertainties will be reduced to 2/3 of their current values. We also assumed that the correlation coefficient would remain the same. In the figure we show the $95\%$ confidence contour.

We also analyzed the broader case with arbitrary $q_f$. Our parametrization (\[novoq\]) allows us to determine the present value of $q$ ($q_0$) in terms of $z_t$, $\tau$ and $q_f$. Although the considered data sets do not impose a lower bound for $q_f$, they do constrain $q_0$ (we found $-1.4\lesssim q_0\lesssim -0.3$). In Fig.\[fig2\] (right-panel) we show, in the parameter space $(z_t,\tau,q_0)$, the $95\%$ confidence surface for the general case ($q_f \in (-\infty,0)$), obtained using SNLS+$S_{k}/D_{v}$ data.

Conclusion
==========

In this work, with a formulation that avoids strong assumptions about the dark sector and/or the metric theory of gravity, we showed that by using only SN data the transition redshift (from decelerated to accelerated expansion) could be very large ($z_t>10$). We demonstrated the importance of combining the SN test with the $S_k/D_v$ test to better constrain the parameters $z_t$ and $\tau$. We introduced the parameter $\tau$ and showed its relevance to characterize models like flat $\Lambda$CDM, DGP and others. We confirmed that there is a tension between *Gold182* and SNLS data sets with a quite general formulation. We also exhibited what should be expected from future $SN + S_{k}/D_{v}$ observations and, relaxing the condition $q_f = -1$, obtained current constraints on the parameters $z_t$, $\tau$ and $q_0$. A more detailed analysis of the consequences of our parametrization in the case $q_f \neq -1$ is still necessary. For instance, our parametrization is not able to describe, in all their redshift range, models that are now accelerating but that decelerates again in the future [@frieman]. The description of the expansion history of these models is more complicated since it requires a second transition. It would be interesting to investigate under what conditions it would be possible to describe, with our parametrization, the behavior of these kind of models for $z>0$. The case $q_i \neq 1/2$ should also be further investigated. These issues will be discussed in subsequent work.

**Acknowledgements** {#acknowledgements .unnumbered}
====================

We thank Maurício Calvão, Sérgio Jorás, Miguel Quartin and Rogério Rosenfeld for useful discussions and suggestions. EEOI, AVT and IW are partially supported by the Brazilian research agency CNPq. RRRR is partially supported by the Brazilian research agency CAPES.

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