# Energy conditions bounds and supernovae data

###### Abstract

The energy conditions play an important role in the description
of some important properties of the Universe, including the current
accelerating expansion phase and the possible recent phase of
super-acceleration.
In a recent work we have provided a detailed study of the energy
conditions for the recent past by deriving bounds from energy
conditions and by making the confrontation of the bounds with
supernovae data. Here, we extend and update these results in two different
ways. First, by carrying out a new statistical analysis for
estimates needed for the confrontation between the bounds and supernovae
data. Second, by providing a new picture of the energy conditions
fulfillment and violation in the light of the recently compiled
*Union* set of type Ia supernovae and by using two
different statistical approaches.

###### pacs:

98.80.Es, 98.80.-k, 98.80.Jk## I Introduction

In the study of the classical energy conditions EC-basics_refs in
cosmological context, an important viewpoint is the confrontation of
their predictions with the observational data.
Since the pioneering papers by Visser M_Visser1997 a number of
articles have been published concerning this confrontation by using
model-independent energy-conditions *integrated* bounds on the
cosmological observables such as the distance modulus and lookback
time Santos2006 – Lima2008
(see also the related Refs. EnergCond_rel ).
Energy conditions constraints on modified gravity models, such as the
so-called –gravity, have also been investigated in
Ref. Santiago2006 and more recently in Ref. SARC2007 .

In a recent work Lima2008 , we have shown that the fulfillment
(or the violation) of these *integrated* bounds at a given redshift
is not sufficient (nor necessary) to ensure the fulfillment (or the violation)
of the energy conditions at . This amount to saying that the local
confrontation between the prediction of the *integrated* bounds
and observational data is not sufficient to draw conclusions on the fulfillment
(or violation) of the energy conditions at .
This crucial drawback in the confrontation between *integrated* bounds
and cosmological data has been overcome in Ref. Lima2008 , where
new *non-integrated* bounds have been derived, and confronted
with type Ia supernovae (SNe Ia) data of the *gold* Riess2007 and
*combined* Combined samples.

In this letter, to proceed further with the investigation of the
interrelation between energy conditions on scales relevant for cosmology
and observational data, we extend and update the results of
Ref. Lima2008 in two different ways. First, carry out a
new statistical analysis for estimates necessary
for the confrontation between the *non-integrated* bounds and
supernovae data. Second, we give a new picture of the energy conditions
fulfillment and violation for recent past ()
by using the recently compiled *Union* sample Kowalski2008
with type Ia supernovae along with the new as well as the
previous Lima2008
statistical tools.

## Ii Non-integrated bounds from the energy conditions

In order to use the energy conditions on cosmological scale, we consider the standard cosmological approach in which the Universe is modelled by a dimensional space-time manifold endowed with a locally homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) metric

(1) |

where the spatial curvature or and is the scale factor. We additionally assume that the large scale structure of the Universe is determined by the gravitational interaction, and hence can be described by the General Relativity theory. These assumptions restrict the energy-momentum tensor to that of a perfect fluid of density and pressure , i.e., . In this context, the energy conditions take the following forms: EC-basics_refs

(2) |

where NEC, WEC, SEC and DEC correspond, respectively, to the null, weak, strong, and dominant energy conditions, and the density and pressure of the cosmological fluid are given by

(3) | |||||

(4) |

where overdots denote the derivative with respect to the time and is the Newton’s gravitational constant.

The *non-integrated* bounds from energy conditions Lima2008
can then be obtained in terms of the deceleration parameter
, the normalized Hubble function
, and the curvature density parameter
, simply by substituting
Eqs. (3) into Eqs. (2) to give

NEC | (5) | ||||

WEC | (6) | ||||

SEC | (7) | ||||

DEC | (8) |

where is the redshift, , and the subscript 0 stands for present-day quantities. Here and in what follows we have used the notation of Ref. Lima2008 , in which NEC, WEC, SEC and DEC correspond, respectively, to , , and .

We note that for a given spatial curvature , NEC [Eq.(5)] and DEC [Eq. (8)] provide, respectively, a lower and an upper bound on the plane for any fixed redshift . The WEC bound [Eq.(6)] only restricts the normalized Hubble function for a fixed value of , while the SEC bound [Eq.(7)] does not depend on the value of the spatial curvature. Thus, for any given value of , having estimates of and for different redshifts , one can test the fulfillment or violation of the energy conditions at each .

In this work, we focus on the FLRW flat () universe, in which the NEC and DEC bounds reduce, respectively, to and . Now, the and estimates are obtained by using a SNe Ia dataset, through a model-independent approach which consists in approximating the deceleration parameter function in terms of the following linear piecewise continuous function (linear spline):

(9) |

where the subscript means that the quantity is taken at , , and the prime denotes the derivative with respect to . The supernovae observations provide the redshifts and distance modulus

(10) | |||||

where , respectively. Then, by using the following relation between and : for

(11) |

along with Eq. (10), we fitted the parameters of the ,
as given by (9), by using the SNe Ia redshift–distance
modulus data from the so-called *Union* sample as compiled by
Kowalski *et al* Kowalski2008 .

## Iii Results and Discussions

We have used two different statistical approaches to confront the energy
conditions bounds with observational data. In the first approach, which
holds only for the flat case^{1}^{1}1Note that in this case the NEC, SEC, and
DEC bounds are independent of .,
we have computed the estimates by marginalizing over
and the other parameters (’s) of the
function, with confidence levels (C.L.).
In the second procedure (which was used in Ref. Lima2008 ), we
have calculated the confidence regions on the
plane, and used the upper and lower
limits of to have the C.L. of
for all (recent past).

To obtain a global picture of the violation and fulfillment of the energy conditions in the recent past by using the first statistical method, we have calculated the estimates at equally spaced redshifts in the interval , and our result are depicted in Fig. 1a which shows the NEC, SEC, and DEC bounds along with the best-fit values and the , and limits of in the plane. We note that WEC bound is fulfilled identically in the flat case.

When an observational confrontation is needed for a non-flat FLRW case () the second statistical procedure has to be employed, because in this case the NEC [Eq. (5)] and DEC [Eq. (8)] bounds depend on the estimates of . In this way, estimates cannot be obtained by marginalizing over , and one has to calculate the confidence regions on the plane (second approach). In this work, however, we have used this approach also for the flat case (which we have focussed on) in order to make a comparison of the observational SNe confrontations obtained by using both statistical procedures, and also with the results of Ref. Lima2008 . Figure 1(b) contains the result of our analysis obtained by using the second procedure. Besides the NEC, SEC, and DEC bounds, it shows the best-fit values, the upper and lower limits of from the confidence regions on the plane calculated for each of previously used equally spaced redshifts in the interval .

The two panels in Fig. 1 show the violation of the SEC with more than in the redshift intervals [panel (a)] and [panel (b)], where the highest evidence of SEC violation is at with [panel (a)] and [panel (b)]. Unlike the result from the confidence regions approach displayed in panel (b), which indicates the breakdown of SEC within in the whole redshift interval, we note that from panel (a) one has that, within , the SEC is fulfilled for and is violated for . This indicates that, within C.L., the Universe crosses over from a decelerated to an accelerated expansion phase for a redshift within the interval .

Regarding the NEC, Fig. 1 indicates its breakdown within for low redshift intervals, i.e., [panel (a)] and [panel (b)]. For higher values of redshifit, NEC is violated with for [see panel (a)] and for [cf. panel (b)].

Concerning the DEC, we note that its fulfillment takes place in most of the redshift interval for both statistical analyses [see panel (a) and panel (b)], but it is violated within for [panel (a)] and [panel (b)], which are intervals where the error bars of our estimates grow significantly.

The comparison of the results obtained through the confidence regions
approach by using the *Union* set of SNe Ia
with those of Ref. Lima2008 calculated through the same
statistical procedure but by employing the and the supernovae
of the *gold* and *combined* samples, shows that
the errors bars of from the Union sample analysis are
smaller for redshifts lying in . The Union sample
results reinforce the indication of the SEC
violation and NEC fulfillment at low redshift pointed out
recently in Ref. Lima2008 .

Here, similarly to the analyses of Ref. Lima2008 , we have found that the results of the analyses for the best fit, the upper and lower values of as given by five-year WMAP Komatsu2008 , are essentially the same of the flat case, with differences much smaller than the associated errors. We note that for the upper limit of , the WEC bound is fulfilled with confidence level in the redshift interval , while for the interval the WEC is identically satisfied.

## Iv Concluding Remarks

In a previous work Lima2008 we provided a picture
of the violation and fulfillment of the energy conditions
in the recent past by deriving *non-integrated*
bounds from energy conditions in terms of the deceleration parameter
and the normalized Hubble function in the context of FLRW cosmology,
and made the confrontation of the bounds with SNe Ia data through
estimates of from confidence regions on
the plane calculated with *gold* Riess2007
and *combined* Combined samples.

Here, we have extended and updated the confrontation between the
*non-integrated* bounds and supernovae data. First, by
using the fact that, in the flat case, the NEC, SEC,
and DEC bounds do not dependent on , we have
carried out a new statistical analysis in which the estimates
are obtained by marginalizing over the parameters along with
the ’s of [see Eq. (9)]. Second, we have
updated the previous work Lima2008 by providing a new picture
[see Fig. 1(a) and [Fig. 1(b)] of the energy conditions
fulfillment and violation from the recently compiled Union set of
SNe Ia along with two different statistical tools.

On general grounds, our analyses indicate a possible recent phase of super-acceleration in which the NEC is violated within confidence level for [Fig. 1(a)] and [Fig. 1(b)], and that the DEC is fulfilled with in the redshift interval [Fig. 1(a)] and [Fig. 1(b)]. Regarding the SEC, our analyses show that, for both statistical approaches employed, the best-fit curve of crosses the SEC–bound curve at , and that SEC is violated with within small low redshift intervals [Fig. 1(a) and [Fig. 1(b)].

Finally, an interesting fact that comes out of our SEC
analysis with C.L., obtained by using the recent SNe Ia *Union*
set, is that for the new estimate [calculated by marginalizing
over ] the deceleration to acceleration transition expansion phase
of the universe took place in the redshift interval ().

## Acknowledgments

This work is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) - Brasil, under grant No. 472436/2007-4. M.P.L., S.V. and M.J.R. thank CNPq for the grants under which this work was carried out. We also thank A.F.F. Teixeira for the reading of the manuscript and indication of the relevant misprints and omissions.

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