#### Gamma-ray bursts as dark energy-matter probes in the context of the generalized Chaplygin gas model

C 2005 The Authors. Journal compilation C 2005 RAS
Gamma-ray bursts as dark energy-matter probes in the context of the generalized Chaplygin gas model
O. Bertolami 0
P. T. Silva 0
0 Instituto Superior Te ́cnico, Departamento de F ́ısica, Avenida Rovisco Pais , 1, 1049-001, Lisboa , Portugal
A B S T R A C T In this paper we consider the use of gamma-ray bursts (GRBs) as distance markers to study the unification of dark energy and dark matter in the context of the so-called generalized Chaplygin gas (GCG) model. We consider that the GRB luminosity may be estimated from its variability and time-lag, and we also use the so-called Ghirlanda relation. We evaluate the improvements expected once more GRBs and their redshift become available. We show that although GRBs allow for extending the Hubble diagram to higher redshifts, its use as a dark energy probe is limited when compared to Type Ia supernovae. We find that the information from GRBs can provide some bounds on the amount of dark matter and dark energy independently of the equation of state. This is particularly evident for XCDM-type models, which are, for low redshifts (z 2), degenerate with the GCG.
methods; miscellaneous - cosmological parameters - cosmology; observations - dark matter - distance scale - gamma-rays; bursts
1 I N T R O D U C T I O N
The generalized Chaplygin gas (GCG) model (Kamenshchik,
Moschella & Pasquier 2001; Bento, Bertolami & Sen 2002b) is
an interesting alternative to more conventional approaches for
explaining the observed accelerated expansion of the Universe such as
a cosmological constant (see, for example, Bento & Bertolami 1999;
Bento, Bertolami & Silva 2001) or quintessence (Ratra & Peebles
1988a,b; Wetterich 1988; Caldwell, Dave & Steinhardt 1998;
Ferreira & Joyce 1998; Amendola 1999; Bine´truy 1999; Chiba 1999;
Uzan 1999; Zlatev, Wang & Steinhardt 1999; Albrecht & Skordis
2000; Bertolami & Martins 2000; Kim 2000; Sen, Sen & Sethi 2001;
Sen & Sen 2001; Bento, Bertolami & Santos 2002a). It is worth
remarking that quintessence is related to the idea that the cosmological
term could evolve (Bronstein 1933; Bertolami 1986a,b; Ozer & Taha
1987) and to attempts to solve the cosmological constant problem.
In the GCG approach, an exotic equation of state is considered to
describe the behaviour of the background fluid
where A and α are positive constants. The case α = 1 corresponds
to the Chaplygin gas. In most phenomenological studies the range
0 < α 1 is considered. Within the framework of Friedmann–
Robertson–Walker cosmology, after being inserted into the
relativistic energy conservation equation, this equation of state leads to
where a is the scale factor of the Universe and B is a positive
integration constant. From this result, we can understand a striking
property of the GCG: at early times the energy density behaves as
matter while at late times it behaves like a cosmological constant.
This behaviour suggests the interpretation of the GCG model as an
entangled mixture of dark matter and dark energy.
This model has several attractive features. From a theoretical point
of view, the pure Chaplygin model (α = 1) equation of state can be
obtained from the Nambu–Goto action for d-branes moving in a
(d + 2)-dimensional space–time in the light cone parametrization
(Bordemann & Hoppe 1993). It is also the only fluid which admits
a supersymmetric generalization (Jackiw & Polychronakos 2000),
and also appeared in the study of the stabilization of branes in bulks
with a black hole geometry (Kamenshchik, Moschella & Pasquier
2000). The Chaplygin gas may be viewed as a quintessence field
with a suitable potential (Kamenshchik et al. 2001), or as an effect
arising from the embedding of a (3 + 1)-dimensional brane in a
(4 + 1)-dimensional bulk (Bilic, Tupper & Viollier 2002). The
generalized Chaplygin gas also has a connection with brane theories
(Bento et al. 2002b). The model can be yet viewed as the simplest
model within the family of tachyon cosmological models (Frolov,
Kofman & Starobinsky 2002).
The GCG model has also been successfully confronted with
different classes of phenomenological tests: high precision cosmic
microwave background radiation data (Amendola et al. 2003; Bento,
Bertolami & Sen 2003a,b,c; Carturan & Finelli 2003), supernova
(SN) data (Fabris, Gonc¸alves & de Souza 2002a; Alcaniz, Jain
& Dev 2003; Dev, Alcaniz & Jain 2003; Gorini, Kamenshchik &
Moschella 2003; Makler, de Oliveira & Waga 2003; Bertolami et al.
2004; Zhu 2004; Bento et al. 2005) and gravitational lensing (Silva
& Bertolami 2003; Dev, Jain & Alcaniz 2004). More recently, it has
been shown using the latest SN data (Tonry et. al. 2003; Barris et al.
2004; Riess et al. 2004), that the GCG model is degenerate with a
dark energy model with a phantom-like equation of state (Bertolami
et al. 2004; Bento et al. 2005). Furthermore, it can be shown that
this does not involve any violation of the dominant energy
condition and hence does not lead to the big rip singularity in future
(Bertolami et al. 2004). It is a feature of GCG model that it can mimic
a phantom-like equation of state, but without any kind of
pathologies as asymptotically the GCG approaches a well-behaved de Sitter
universe. Structure formation in the context of the Chaplygin gas
and the GCG was originally examined in Bento et al. (2002b), Bilic
et al. (2002) and Fabris, Gonc¸alves & de Souza (2002b). The
results of the various phenomenological tests on the GCG model are
summarized in Bertolami (2004, 2005),
Subsequently, concerns about such a unified model were raised
in the context of structure formation. Indeed, it has been pointed
out that unphysical oscillations should be expected or even an
exponential blow-up in the matter power spectrum (Sandvik et al. 2004),
given the behaviour of the sound velocity through the GCG.
Although, at early times, the GCG behaves like dark matter and its
sound velocity is vanishingly small, at later times the GCG starts
behaving like dark energy with a substantial negative pressure,
yielding a large sound velocity which, in turn, can produce oscillations
or blow-up in the power spectrum. This is a common feature of any
unified approach when the dark matter and the dark energy
components of the fluid are not clearly identified. These components are,
of course, interacting, as they make part of the same fluid. It can be
shown however that the GCG is a unique mixture of interacting dark
matter and a cosmological constant-like dark energy, once the
possibility of phantom-type dark energy is excluded (Bento, Bertolami &
Sen 2004). It can be shown that as a result of the interaction between
the components, there is a flow of energy from dark matter to dark
energy. This energy transfer is not significant until the recent past,
resulting in a negligible contribution at the time of gravitational
collapse (z c 10). Subsequently, at about z 0.2, the interaction starts
to grow, yielding a large energy transfer from dark matter to dark
energy, which leads to the dominance of the latter at present. Actually,
it is shown that the epoch of dark energy dominance occurs when
dark matter perturbations start deviating from its linear behaviour
and that the Newtonian equations for small-scale perturbations for
dark matter do not involve any mode-dependent term. Thus, neither
oscillations nor blow-up in the power spectrum do develop.
In this paper we study yet another cosmological test and its
possible use to study the GCG. Schaefer (2003) suggested that
gammaray bursts (GRBs) may be used to extend the Hubble diagram to
redshifts as high as z ∼ 5. For ‘ordinary’ dark energy, such high
redshifts are not very interesting because at those epochs the
Universe is dominated by dark matter, and thus it is less sensitive to the
nature of dark energy.
For the GCG however, the GRB test might be relevant because it
unifies dark energy and dark matter into one single fluid. Therefore,
within the framework of the GCG model, the dark matter domination
period actually depends of the nature of the dark energy component
that kicks in at later times, and we can expect that the study of the
matter-dominated era will bring some insight into some properties
of GCG models.
This paper is organized as follows. In Section 2 we explain our
method of using the time-lag/luminosity and variability/luminosity
correlations to constrain cosmological models. In Section 3 we
present and comment on the results obtain from this method. In
Section 4 we consider a more precise correlation found by Ghirlanda,
Ghisellini & Lazzati (2004a), and study its consequences. In
Section 5 we consider whether it is possible to use the extended redshift
range of GRBs to break the degeneracy between the GCG and the
XCDM (cold dark matter plus a dark energy component with a
constant equation of state). Finally, in Section 6 we discuss our results
and present our conclusions.
2 M E T H O D
2.1 Overview
The starting point of our study is the proposed correlation between
time-lags in GRBs spectra and the isotropic equivalent
luminosity (Norris, Marani & Bonnell 2000), and the correlation between
GRB variability and isotropic equivalent luminosity (Reichart et al.
2001). The time-lag, denoted by τ lag, measures the time offset
between high- and low-energy GRB photons that arrive on Earth. The
variability, V, is easily defined in qualitative terms as a
measurement of the ‘spikiness’ or complexity of the GRB light curve. The
isotropic equivalent luminosity is the inferred luminosity of a GRB
if all its energy is radiated isotropically. That is, if P is the peak
flux of a burst in units of photons cm−2 s−1 between observer frame
energies El and Eu, the isotropic equivalent peak photon
luminosity of the burst in erg s−1 in the source frame energy range 30 to
2000 keV is given by
where N(E) is the source frame spectral shape, usually parametrized
by a Band function (Band et al. 1993), and r(z) is the comoving
distance to a burst at redshift z.
The possible use of this relation to expand the Hubble diagram
to higher redshifts was first discussed in Schaefer (2003). One
limitation of the employed method is the cosmological distances of
GRBs, which affect the ability of performing a proper calibration
of their distance independently of the background cosmology. It is
necessary either to fix a cosmological model and find a calibration
that depends on the assumed cosmological model, or to fit the data
to both calibration and cosmological parameters. This degrades
precision because there are more free parameters for the same number
of data points.
One way around this was proposed by Takahashi et al. (2003).
Let us assume that we measure the luminosity distance up to z =
z max, with say z max = 1.5. This is, for instance, very likely to be
possible with the Supernova Acceleration Probe (SNAP)1 mission.
This means that we would have an estimate of the absolute isotropic
luminosity that is independent of the calibration and of the
cosmological model. We can then use these estimates to calibrate the
(τ lag, L iso) and (V , L iso) relations without assuming a background
cosmological model.
The major strength of GRBs as cosmological probes is that they
can be found at very high redshifts (Lamb & Reichart 2000). We
can then use high-redshift GRB data to probe the cosmology. This is
1 See http://snap.lbl.gov/.
done by using the (τ lag, L iso) and (V , L iso) relations to estimate the
luminosity distance at higher redshifts. A possible method would
be to use GRBs with z < 1.5 together with the luminosity distance
estimates from SNe to calibrate the luminosity estimator, and then
use this calibration to find the luminosity distance of GRBs with z >
1.5. It should be noted that this method aims to study the luminosity
distance in the range 1.5 < z < 5. Later, it will be shown that adding
information obtained in the range z < 1.5 is actually crucial to the
study of dark energy models.
With this information, we can estimate the luminosity distance at
high redshifts, and place constraints on the cosmological parameters
via a standard χ 2 minimization procedure.
The aim of this paper is to use this method to constrain the GCG
unification model of dark energy and dark matter. Even though it is
shown that the optimum redshift range for studying dark energy is
around z < 2 (Huterer & Turner 2001), as already remarked, because
the GCG also describes dark matter, a higher redshift range might
be relevant for a better understanding of the model.
Our study is performed in three steps. First, we build a realistic
mock distribution of GRBs in redshift and isotropic luminosity space
(Section 2.2). Secondly, we test the calibration procedure to find
what improvements might be achieved in the future. To do this, we
consider a fiducial set of calibration parameters to generate a mock
set of time-lags and variabilities for each GRB. We then perform
a χ 2 fit to these mock data to study the calibration precision. The
last step consists of employing this method to probe GCG models.
This is done in a fashion similar to what was already performed with
SNe Ia (Goliath et al. 2001; Weller & Albrecht 2002; Di Pietro &
Claeskens 2003; Silva & Bertolami 2003). A fiducial cosmological
model is assumed, and regions of constant χ 2 are plotted around the
fiducial set of parameters.
2.2 Generating a gamma-ray burst mock population
We describe here the process of population generation. We
simulate several data sets, which differ only in size. We consider three
sample sizes. The first sample is composed of 150 GRBs, and is
consistent with the expected number of GRBs with measured
redshift z > 1.5, which is what the Swift satellite is expected to detect
in its three-year mission. The second and third samples are larger,
containing 500 and 1000 GRBs, respectively, and serve as best-case
scenarios to test whether further data might improve the results. The
GRB distribution on redshift will be part of the data generated by
future GRBs surveys. Thus, our mock catalogues do not need to be
extremely precise; however, they should be realistic.
The GRBs in each sample are distributed in redshift and
luminosity according to the GRB rate history and luminosity function
based on the model of the star formation 2 from Porciani & Madau
(2001).
Denoting by ψ (L) the GRB luminosity function, normalized to
unity, then the observed rate of bursts with observed peak flux larger
than P1 at the redshift range (z, z + dz) is given by
P1) = dz
dV (z) RGRB(z)
Here RGRB is the comoving GRB rate density, ( P) is the
detector efficiency as a function of photon flux, which for simplicity is
assumed to be ( P) = 1, and L( P, z) is given by equation (3).
The GRB rate is calculated assuming that it tracks the global star
formation history (Porciani & Madau 2001), i.e. R GRB ∝ R SF ∝
R SN, where RSF and RSN are the comoving rate densities of star
formation and core collapse (Type II) SNe.
We use the following parametrization (Steidel et al. 1999;
Porciani & Madau 2001)
yr−1 Mpc−3,
exp(3.4z)
RSF(z) = 0.15h65 exp(3.4z) + 22 M
where h65 is the Hubble constant in units of 65 km s−1 Mpc−1. These
numerical factors were calculated for an Einstein–de Sitter universe.
To convert this result into the CDM model, a factor of H (z| m,
, h 65)/H (z|1, 0, 1) must be introduced as a multiplication factor;
here H(z) is the Hubble factor as a function of redshift (see appendix
A of Porciani & Madau 2001 for details).
To calculate the rate density of SNe II, it is assumed that all
stars with masses M > 8 M explode as core-collapsed SNe. This
yields an additional factor of 0.0122 M−1 to be multiplied to the star
formation rate. Finally, using the results obtained from Burst and
Transient Source Experiment (BATSE) bursts (Porciani & Madau
2001), we have
0.0122h265 H (z| m,
RGRB(z) = 4.4 × 105
We use the luminosity function (Porciani & Madau 2001)
Here, L denotes the rest-frame peak luminosity in the 30–2000 keV
energy range, and C = [L 0 (−γ − 1)]−1, where (x ) is the Euler
gamma function, ensures a proper normalization. Using the results
from Porciani & Madau (2001) obtained for a GRB formation rate
given by equation (6), we consider γ = −2.9, L 0 = 7 × 1051 h6−52
erg s−1
To calculate L( P, z) we need to know the photon energy spectrum.
For simplicity, we describe the energy spectrum of the GRB as a
single power law with index −2.5, instead of the customary Band
function (Band et al. 1993). To check whether this choice affects
our results, we have calculated some mock catalogues using the
Band function used in Porciani & Madau (2001). The only effect we
have found was a small increase in the number of detected GRBs
at very high (z > 5) redshifts. Because the measurement of such
large redshifts is unlikely (see however Lamb & Reichart 2000), we
placed an upper limit of z 5 when building our samples. Thus,
our results are not altered by using the Band function or the power
law.
The flux limit of the Swift satellite, P > 0.04 photons cm−2 s−1,
is applied to check whether the GRB can be detected. The
observed magnitude is calculated assuming a flat CDM cosmological
model, with = 0.7 and H 0 = 70 km s−1 Mpc−1.
Because equation (4) gives us the number of GRBs that are
expected to be detected in the redshift range (z, z + dz), we have that
the function
∞ dN
φ(z, P > P1) = dN (z, P > P1)
0
gives the probability of detecting a GRB in that redshift range. With
this probability density function, we built a pseudo-random number
generator to generate our mock catalogues (see, for instance, Press
et al. 1992).
We show an example of a generated sample in Fig. 1.
(z, P > P1) dz
−1
2.3 Calibration procedure
Reichart et al. (2001) proposed a relation between the variability
and isotropic equivalent luminosity of a GRB such that
Liso = Bτ τlβagv . (10)
As already mentioned, the first step in testing the calibration
procedure consists of establishing a fiducial model. Schaefer (2003)
used the nine GRBs with available redshifts at the time to calibrate
these relations, yielding
Bv = 1050.03,
We assume these values as our fiducial model, i.e. we suppose that
the calibration relations are faithful, and that they are described
by this set of parameters. For each GRB of the mock luminosity
distribution, generated as we have explained above, we compute the
corresponding time-lag, τ lag, and variability, V, using
1
log τlag = log Bτ + βτ log L + Random(στ ), (13)
1
log V = log Bv + βv log L + Random(σv),
where the Random (σ ) term is a pseudo-random number drawn from
a normal distribution with zero mean value and variance σ . That is,
we assume that τ lag and V have lognormal error distributions, with
variance σ τ and σ v, respectively.
The values we use for σv and στ are based on Schaefer (2003)
and are essentially dominated by an intrinsic (statistical) error. The
used fiducial values are shown in Table 1.
In the real situation, we would assume that the luminosity distance
at redshifts smaller than z = 1.5 has been measured independently
by SNe Ia experiments, such as SNAP. Knowledge of the peak flux
observed on Earth and the spectral shape N(E) allow us to obtain the
corresponding equivalent isotropic luminosity from equation (3).
We do not consider errors in this estimate of Liso, but it should
be noted that it is expected that the luminosity distance uncertainty
will be around σ log dL ∼ 0.01 for future experiments such as SNAP
(Goliath et al. 2001; Weller & Albrecht 2002). This translates into
σ log L ∼ 0.02, a much smaller error than the intrinsic scatter of
GRBs. Thus, this source of uncertainty should not influence our
conclusions.
Notice that we have used low-redshift GRBs from the generated
sample to calibrate the relations equations (9) and (10) through a
standard χ 2 fitting procedure. That is, our procedure consisted the
following:
(i) generating a GRB sample;
N GRB used in calibration
(ii) for each GRB in the sample, assigning a time-lag and
variability via equations (13) and (14);
(iii) fitting the calibration relations equations (9) and (10) for the
low-z GRBs from the mock sample;
(iv) repeating the steps for different size samples, and studying
the precision of the attained fits.
Our results are shown in Table 2. The smaller sample size
corresponds to what is expected to be found by the Swift satellite in
its three-year campaign; the second sample size corresponds to the
number of low-z GRBs that should be found if there were about
500 GRBs in total, while the third sample corresponds to about
1000 GRBs in total. It should be noted that the smaller sample
represents a fourfold improvement on the sample used by Schaefer
(2003).
2.4 Estimating Liso and the χ2 test
For the remaining GRBs that were not used in the calibration
procedure, the calibrated relations are used to find Liso from the values
of τ and V. Thus, we obtain two estimates:
log L V = log Bv + βv log V ,
log Lτ = log Bτ + βτ log τlag.
These two estimates are combined as weighted averages to produce
one estimate of Liso:
mi − Mi − 25 − 5 log dL (zi , p) 2
where mi is the observed magnitude, Mi is the absolute magnitude
estimated from equations (15) and (16), and dL(zi, p) is the standard
luminosity distance as function of redshift z and the cosmological
parameters p. The denominator σµ is the uncertainty in the
determination of the distance modulus, µ = mi − Mi. This uncertainty
was calculated using Gaussian error propagation, and assuming that
there were no correlation terms to be taken into account. The values
of σµ we have found are shown in Table 2. We have to minimize this
function and draw the χ 2 contours in order to find the confidence
regions.
Here we follow the method developed for the study of SNe Ia
(Goliath et al. 2001; Weller & Albrecht 2002; Di Pietro & Claeskens
2003; Silva & Bertolami 2003). We use the log-likelihood function
χ 2 to build confidence regions in the parameter space. In order to
perform this, we choose a fiducial model, denoted by the parameter
vector pfid, and then the log-likelihood functions χ 2 are calculated
based on hypothetical magnitude measurements at the various
redshifts. The χ 2 function is then given by
5 log dL (zi , pfid) − 5 log dL (zi , p) 2
Of course, there is no need to minimize the χ 2 function because its
minimum will correspond to p = pfid; thus, we only need to find the
χ 2 contours corresponding to the desired confidence level (CL).
2.5 The models
The GCG model smoothly interpolates between a dark matter
dominated time in the past, to an accelerated de Sitter phase in the future.
Thus, this is a setting that on large scales agrees with the observed
expansion history of the Universe (Kamenshchik et al. 2001; Bento
et al. 2002b). The GCG density may be written as a function of
redshift as
ρch(z) = ρch,0 As + (1 − As)(1 + z)3(1+α) 1/(1+α)
( A + B)1/(1+α) = ρch,0,
where B is the integration constant that appears in equation (2). For
z 0 we have a matter-dominated universe
0) = (1 − As)1/(1+α)(1 + z)3,
while in the far future, z = −1, the GCG behaves as a
vacuumdominated universe.
The GCG unifies dark matter and dark energy, but does not take
into account the presence of baryons and radiation. The baryonic
component has to be considered when studying the implications of
the GCG model with regard to observational tests, but for geometric
tests, such as the magnitude-redshift relation we consider here, the
effect of these components is negligible. Therefore, throughout the
paper we disregard the presence of baryons and radiation. This does
not affect any of our conclusions.
For the purpose of comparison we choose the so-called XCDM
model, which consists of two components, cold dark matter and
some form of dark energy which has a constant negative equation
of state, w = p/ρ. We use this parametrization to test GRBs as
a probe of an unspecified dark energy component. Although this
parametrization is not suitable for general dark energy models,
because in most cases the equation of state changes with time, it is
adequate to test the cosmological constant model, w = −1, against
other models. If it is found that w = −1 is disfavoured by the data,
then there is a strong indication that the dark energy component is
more complex than expected.
For many years cosmologists were mainly concerned with
models to which −1 w < −2/3; however, more recently mounting
evidence that w < −1, the so-called phantom dark energy models
(Caldwell 2002; Carrol, Hoffman & Trodden 2003), is being
encountered. In many models there are several theoretical reasons not
to consider w < −1, most notably that this would lead to a
breakdown of the weak, the dominant and the strong energy conditions
(ρ 0 and ρ + p 0, ρ | p|, and ρ + 3 p 0, respectively).
The weak and strong energy conditions are those to hold in
proving well-known singularity theorems, while the dominant energy
condition guarantees the stability of a component, such that there is
no creation of energy-momentum from nothing (Hawking & Ellis
1973).
Another feature of some phantom models is that they exhibit a
future blow-up of the scale factor in a finite time, often referred to as
the ‘big rip’. Because such a universe has a finite lifetime, it has been
argued that these phantom models solve the coincidence problem;
that is, they explain why dark matter and dark energy densities are
of the same order at the present time. For a CDM universe, a long
matter-dominated period is followed by a rather quick transition to a
phase such as the one the Universe is in now, where matter and dark
energy have approximate densities. This phase is then followed by
an eternal vacuum-dominated exponential expansion. Within this
framework, the probability of finding ourselves in this intermediate
and temporary epoch is very small. For phantom models, however,
the accelerated expansion is not eternal. Because the Universe has a
finite lifetime, the probability of living in the epoch of matter-energy
approximate equality is larger than that of CDM models. Thus,
the cosmic coincidence is not as unlikely as for CDM models.
However, these models fail to explain why the dark energy
component did not start to dominate the evolution at an earlier time, i.e.
why dark energy started dominating the cosmic evolution only after
the large-scale structure had time to evolve deep into the non-linear
regime, even though in the context of the CGC this is exactly what
is found (Bento et al. 2004).
Furthermore, besides these theoretical features, the latest SNe
Ia data do seem to favour a phantom energy component. Thus, in
what follows, we consider a phantom model for comparison with
the GCG model. Moreover, as shown in Bertolami et al. (2004) and
Bento et al. (2005), for z < 2, the GCG model is degenerate with
phantom models with suitable parameters.
As mentioned, we consider two different cosmological models:
the flat GCG model, which unifies dark energy and dark matter into a
single component, and the XCDM, which parametrizes dark energy
in terms of a constant equation of state, w = p/ρ. We consider
that the Universe is flat in both models. The GCG model is then
described by the exponent α, and the quantity As, while the XCDM
is described by the parameters w and m, the non-relativistic dark
matter density relative to the critical one. We have considered two
fiducial models. Model I assumes that the Universe is described by
a GCG model with 1 − As = 0.3 and α = 1. Model II assumes the
corresponding degenerate Universe, described by a phantom XCDM
model, with w = −1.4 and m = 0.45.
3 R E S U LT S
In Table 2 we show the estimated uncertainty in the distance modulus
for each calibration. As can be observed, calibrating the luminosity
estimators with just 40 low-redshift GRBs will yield a satisfactory
estimate of the distance modulus. This uncertainty of σµ ≈ 0.68 is
close to half the uncertainty found by Schaefer (2003), who used
only nine GRBs to perform the calibration. Because we are dealing
with a population that shows a large statistical uncertainty,
increasing the number of GRBs used in the calibration will be rewarded
by a smaller systematic error on the distance modulus. However,
as its value is decreased, the systematic contribution to the distance
modulus uncertainty will be overwhelmed by the statistical scatter
Figure 2. Confidence regions for the GCG model as a function of the
number of GRBs. The curves show the 68 per cent CL regions, from the
outer to the inner curves, corresponding to 150, 500 and 1000 high-redshift
GRBs.
Figure 3. Confidence regions for the XCDM model as a function of the
number of GRBs. The curves show the 68 per cent CL regions, from the
outer to the inner curves, corresponding to 150, 500 and 1000 high-redshift
GRBs.
of the population. This may be seen for the calibrations that were
performed using 100 and 200 GRBs. Although the calibration
precision is improved, there is little effect on σµ.
We then tested the effect of an increase in the number of
highredshift GRBs in the test. We assume for the purpose of calibration
that 100 low-redshift GRBs are known, and use the corresponding
calibration precision shown in Table 2. We next consider larger
samples of GRBs. Samples with 150, 500 and 1000 high-redshift GRBs
are examined, and the respective confidence regions are exhibited.
These results are shown in Figs 2 and 3.
Finally, we study the effect of the distribution of GRBs in redshift
space. One of the main advantages of GRBs is that they probe a
redshift range that is far beyond the possibilities of any other known
standard candle such as SNe Ia. Previously we considered samples
that consisted of only high-redshift GRBs to verify what could be
gained by studying this redshift range exclusively. The next point to
be considered is whether the loss of information at lower redshift is
important. To answer this question we added 100 GRBs with z < 1.5
to a sample of 400 high-redshift GRBs, and found the corresponding
confidence regions. It is assumed that this low-redshift sample was
not used in the calibration of the luminosity indicators, thus avoiding
any circularity problems. The results are shown in Figs 4 and 5.
The first conclusion we can draw is that GRBs are not quite
suitable as stand-alone probes of dark energy. This is seen from the
lack of impact that an increase in the redshift range has on the results
in terms of precision and discriminating power.
The main source of uncertainty comes from the intrinsic
statistical scatter of the GRB population. As may be seen from Table 2,
calibrating the (τ , L iso) and (V , L iso) relations with more than about
40 low-redshift GRBs does not substantially affect the results.
However, an increase in the size of the higher-redshift GRB population
will improve the quality of the constraints that can be imposed.
This can be seen in Figs 2 and 3. Using more precise luminosity
estimators, such as the one we use in the following section, might
improve the outcome, even though better results can be achieved by
probing a greater sample of z < 1.5 redshift sources. This is evident
for XCDM models, as shown in Fig. 5. Adding an additional 100
GRBs with z < 1.5 to a sample of 400 high-redshift GRBs is more
effective than adding an additional 500 GRBs with z > 1.5.
Regarding the results for each model, the first conclusion is that
the XCDM model is better constrained than the CGC model. It
can be seen that no constraint in the α parameter for the GCG
model can be imposed, even though an upper limit for As can be
obtained.
As for XCDM models, we point out that there is some potential
to probe m, but the prospect of using this test to probe the nature of
dark energy is very limited. An upper value for w can be obtained;
however, no lower limit. It should be noted that other tests such as
SNe Ia are capable of imposing tighter constraints on m than those
we find, as we show later.
4 T H E G H I R L A N D A R E L AT I O N
4.1 Description
The (τ , L iso) and (V , L iso) relations are notoriously affected by the
large intrinsic scatter of the data set, which greatly hinders their use
as precision cosmological probes. Quite recently, Ghirlanda et al.
(2004a) have found a surprisingly tight correlation between the peak
energy of the gamma-ray spectrum, Epeak (in the ν–ν Fν plot), and
the collimation corrected energy emitted in gamma-rays, E γ , for
long GRBs. This collimation corrected energy measures the energy
release by the GRB taking into account that the energy is beamed
into a jet with aperture angle θ .
Let us denote by Eiso the isotropically equivalent energy, inferred
from the isotropic GRB emission. This source frame
‘bolometric’ isotropic energy is found by integrating the best-fitting
timeintegrated spectrum N(E) (photons cm−2 keV−1) over the energy
range 1–10 MeV:
1
where E is in keV.
If the gamma-ray emission is collimated into a jet with aperture
θ , then the true energy emitted is E γ = E iso(1 − cos θ ). Thus, to
Eiso =
E N (E ) dE ,
convert Eiso into E γ and vice versa, we need to know the angle θ .
Under the simplifying assumption of a constant circum-burst density
medium of number density n, a fireball emitting a fraction ηγ of its
kinetic energy in the prompt gamma-ray phase would show a break
in its afterglow light curve when its bulk Lorentz factor becomes
of the order of 1/θ , with θ given by (Sari 1999)
where tjet is the break time in d, and E iso,52 = E iso/1052 erg.
The Ghirlanda relation (Ghirlanda et al. 2004a; Xu 2005) is then
expressed as
1050 erg = C
where a and C are dimensionless parameters.
Using this result, together with equations (23), (25) and the
definition of E γ , we obtain an equation for dL(z)
Eiso
Note that both Eiso and θ depend on dL(x ), as may be seen from
their definitions. Solving this equation for dL(z) we find an estimate
of the luminosity distance at redshift z in terms of the observables
E p, t jet, n, and fluence S γ .
The error budget for dL is then found to be (Xu 2005)
1 Cθ
where C 2
θ = θ sin θ /(8 − 8 cos θ ).
The distance modulus is given by µobs = 5 log dL/10 pc, and thus
its error is σµ = 5/ln 5 (σ d L/dL). We use the error estimates from
Xu (2005), which are consistent with what is found in the literature
(Ghirlanda et al. 2004a,b; Friedman & Bloom 2005b; Ghisellini
et al. 2005; Mo¨rtsell & Sollerman 2005; Xu, Dai & Liang 2005).
These values yield an uncertainty in µobs of the order of σµ 0.5,
which is just slightly smaller than the error bars from using both the
(L iso, V ) and (L iso, τ L ) relations. As can be observed in Table 3,
the smaller intrinsic scatter of the Ghirlanda relation is balanced
by its dependence on poorly constrained quantities. Improving the
calibration will not solve the problem, because the main sources
of uncertainty are a result of the determination of the peak energy
Ep, the jet break time tjet and the value of the circum-burst density
n. The determination of tjet will become much more precise in the
near future thanks to Swift, but unfortunately the relatively narrow
spectral range of Swift will greatly hinder the determination of Ep
(Friedman & Bloom 2005b). Using additional data from the
HighEnergy Transient Explorer II (HETE II) experiment may reduce this
uncertainty thanks to its larger bandpass of [30, 400] keV (compared
to the [15, 150] keV bandpass of Swift).
Table 3. Error budget for the Ghirlanda test. The uncertainty in the
circumburst density is assumed to be 50 per cent, while the other values are taken
from Xu (2005).
Error
Contribution
to (σ dl /d l)2
Percentage of (σ dl /d l)2
(per cent)
4.2 Results
In Figs 6 and 7 we see the improvements that can be obtained by
the use of the Ghirlanda relation. We have assumed two redshift
distributions: one made up of 500 GRBs with z > 1.5, and the other
made up of 100 GRBs with z < 1.5 plus 400 GRBs with z > 1.5,
as before. For the CGC model we find that the conclusions that can
be drawn remain essentially unchanged. It is obvious that the high
redshifts of GRBs are not quite suitable to adequately study the
GCG model.
Regarding the XCDM model, a rather useful limit to the total
amount of matter may be imposed, but the use of some low-redshift
GRBs is fundamental to constrain the dark energy equation of state.
However, even with these low-redshift GRBs, the uncertainty on w
is still large, especially when compared to SNe Ia.
built the expected confidence regions for SNAP using the method
discussed in Silva & Bertolami (2003) and Bertolami et al. (2004),
and then built joint confidence regions for SNAP and GRBs. We
choose to marginalize over the unwanted parameter M (Goliath
et al. 2001). We considered the use of the Ghirlanda relation on a
sample comprising 500 high-redshift GRBs. The results are shown
in Figs 8 and 9.
Finally, we have studied whether GRBs might improve the results
provided by SNe Ia. Because we are considering future GRB
surveys, we use the SNAP experiment as the SNe Ia benchmark. We
Figure 8. Joint constraints from SNAP plus 500 high-redshift GRBs for
an XCDM model. The dashed line corresponds only to SNAP constraints,
while the solid region corresponds to the SNAP+GRB constraints. All curves
correspond to the 68 per cent CL. Notice that an improvement, although
marginal, is obtained.
The use of GRBs will produce a quite modest improvement
relative to the SNAP result. In fact, we find that the uncertainty on the
distance modulus determined by GRBs would have to decrease by
a factor of 2 or more in order that GRBs improve SNAP allowed
regions noticeably.
In favour of GRBs though, it should be noted that if a large GRB
sample becomes available before the SNAP data, GRBs might
improve SNe Ia constraints, as the SNe constraints will not be as
precise as those expected for SNAP. This is especially true for XCDM
models, where, as we have mentioned above, GRBs are expected
to provide a fair and independent constraint on m. Still, it should
be noted that available SNe Ia data place limits on m that are
close to those predicted here (Tonry et al. 2003; Choudhury &
Padmanabhan 2005). Thus, it seems likely that GRBs will
eventually play more the role of a consistency test than a tool of precision
cosmology.
5 D E G E N E R AC Y B E T W E E N M O D E L S
The CGC and the phantom XCDM models are degenerate at
redshifts z < 2, as shown in Bertolami et al. (2004) and Bento et al.
(2005). Consider the Taylor expansion of the luminosity distance
up to the fourth power of redshift (Visser 2004):
1 1
z + 2 (1 − q0)z2 − 6
1 − q0 − 3q02 + j0 z3
Here, q0 is the deceleration parameter, related to the second
derivative of the expansion factor, j0 is the so-called ‘jerk’ or state finder
parameter (Alam et al. 2003; Sahni et al. 2003), related to the third
derivative of the expansion factor, and k0 is the so-called ‘kerk’
parameter, which is related to the fourth derivative of the
expansion parameter (Dabrowski & Stachowiak 2004; Visser 2004), all
evaluated at present. These quantities are defined as
−2
−3
−4
These are related to each other through the relations
− 1,
j (z) = q(z) + 2q2(z) + (1 + z)
d j
k(z) = −(1 + z) dz (z) − 2 j (z) − 3 j (z)q(z), (35)
where ρ(z) and p(z) refer to the total density and pressure of the
Universe, respectively. With these expressions it is possible to write
the present-day values of the parameters as a function of the
cosmological parameters of the model under examination.
For the XCDM model, we have
m)] − 1,
while for the GCG model we find
q0GCG = 32 (1 − As) − 1,
For the redshift range probed by SNe Ia we can neglect terms
beyond the cubic power in redshift in equation (29). SNe Ia data
indicate that these models have the same deceleration and jerk
(Bertolami et al. 2004; Bento et al. 2005), i.e. they are
degenerate for the considered redshift range. Thus, imposing this equality
we find the relationship between parameters
m =
This degeneracy holds for SNe Ia, for the maximum probed
redshift of about z ≈ 2. As the redshift range allowed by GRBs is greater,
we hope to test higher-order terms in equation (29). At the next
order, the degeneracy is broken by the kerk parameter; that is, even
if q0GCG = q0XCDM, j 0GCG = j 0XCDM, we find that k0GCG = k0XCDM. For
instance, if we consider α = 1 and As = 0.7, then k0GCG = −0.68,
while for the corresponding XCDM model (with w = −1.3 and
m = 0.46), the kerk is k0XCDM = 1.02.
This procedure can be viewed as a consistency test. Indeed,
consider the GCG model. Once the deceleration, the jerk and the kerk
of the Universe are measured, we can extract the values of α and As.
These values yield a value for the kerk, called kt0, which can be
compared to the measured value of the kerk, k0m. Whether both values are
close or not, we can reach a conclusion regarding the suitability of
the model. The same reasoning can be applied to the XCDM model.
This is, of course, a simplification, because we have to consider the
accuracy of measurements and the statistical significance of each of
the parameters in equation (29), as well as considering correlations
between them.
6 D I S C U S S I O N A N D C O N C L U S I O N S
We have verified that the medium and high redshifts of GRBs do
not impose too strong constraints on the GCG model. At such high
redshifts, the expansion factor for the GCG becomes
0) =
ch(1 − As)1/(1+α)(1 + z)3
where ch is the CGC density relative to the critical one. Because
we are considering a flat universe made up mostly by the GCG,
ch = 1. Thus, the luminosity distance at such high redshifts depends
essentially on (1 − As)1/(1+α). It is found that the confidence regions
for the GCG model approximately follow the line (1 − As)1/(1+α) =
const. Hence, because As and α are strongly correlated by this
expression, neither one is constrained.
As for the XCDM model, a high-redshift population of GRBs is
suitable to constrain the total amount of matter, m, but such
highredshift GRBs are poor probes of the dark energy equation of state.
This does not mean that GRBs have no use for the study of dark
energy, or that expanding the Hubble diagram is meaningless. It is
a common trend in the recent literature to argue that a precise prior
estimate of m enhances the ability of SNe Ia to constrain the dark
energy equation of state (Goliath et al. 2001; Weller & Albrecht
2002; Di Pietro & Claeskens 2003). Our results are in complete
agreement with these studies (Dai, Liang & Xu 2004; Ghirlanda
et al. 2004b; Friedman & Bloom 2005a,b; Ghisellini et al. 2005;
Mo¨rtsell & Sollerman 2005; Xu et al. 2005; Xu 2005).
For GRBs, a tight constraint on m is also important to study the
equation of state parameter w. A high-redshift sample will constrain
m, while a low-redshift sample will place constraints on the
equation of state. However, we have verified that a low-redshift sample
alone will fail to constrain the equation of state, because it will be
hindered by large uncertainties in m. This is somewhat similar
to what happens in SNe Ia tests. A low-redshift sample will
constrain the parameter M, while the high-redshift sample constrains
the cosmological parameters (Padmanabhan & Choudhury 2003;
Choudhury & Padmanabhan 2005). Thus, we conclude that while
the use of high-redshift GRBs will allow probing the dark matter
component, to study the nature of dark energy, a low-redshift probe
is necessary. We could of course use GRBs, but for the z < 1.5
range, SNe Ia are the natural choice because of their precision.
Our study also reveals that the effectiveness of the GRB test
depends on the model under examination. GRBs allow us to impose
S− 0.6
O
E
ta− 0.8
l
o
T
− 1
Figure 10. Total equation of state of the Universe as a function of
redshift. The value z = −1 corresponds to the very far future. The dashed line
represents our GCG fiducial model, while the solid line corresponds to the
XCDM phantom model.
some constraints on the XCDM model, but very little can be
extracted for the GCG model. This may be explained by two factors.
The first has been explained above, and is related to the fact that the
region probed by GRBs depends on a unique function of both
parameters. The second is related to how late the Universe stops being
matter-dominated. As shown in Fig. 10, the CGC model remains in
the matter-dominated phase for a longer time, and thus the period
of time during which the dark energy behaviour affects the cosmic
expansion is slightly larger for the XCDM model. Because GRBs
essentially probe larger redshifts, models in which the transition
between the matter-dominated and accelerated expansion is shorter
cannot be well constrained by this test. By the same reasoning,
GRBs might actually be useful in testing models where the
transition between matter-dominated expansion and dark energy driven
acceleration started at larger redshifts. In general, we expect that
GRBs are particularly suited to study models that depart from the
CDM behaviour at larger redshifts.
Thus, we find that despite all intrinsic limitations, larger samples
of GRBs can potentially determine the total amount of matter in
a XCDM model. This independent determination of m will be
useful in studies of the dark energy equation of state, at least until
SNAP becomes a reality. If SNAP provides the scientific bounty it
promises, then GRBs can be regarded as a consistency test. On the
run up to SNAP though, constraints imposed by GRBs may become
quite helpful in studies of dark energy.
Furthermore, GRBs may also be used to break the degeneracy
between models, because at high redshifts the effect of higher-order
terms in the Taylor expansion of dL(z) must be taken into account.
Naturally, a word of caution on the proposed methodology is
in order though, as very little is known about the physics behind
the correlations used. The L iso–τ and L iso–V relations are purely
phenomenological, and no well-established explanation exists,
although some tentative explanations have been put forward (see Ioka
& Nakamura 2001; Plaga 2001; Schaefer 2004). The situation for
the Ghirlanda relation is more complex, as this correlation depends
on poorly constrained quantities, such as the circum-burst density
n, and the gamma-ray creation efficiency, ηγ , and a model of the jet
structure must be assumed. Also, as noted by Levinson & Eichler
(2005), the beaming correction appears to be biased because of the
fact that the inferred opening angle depends on Eiso (see equation
25). Furthermore, several other systematic effects have yet to be
considered, namely selection effects (Band & Preece 2005; Nakar
& Piran 2005) as well as gravitational lensing and other effects
(Friedman & Bloom 2005a).
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