Gamma-ray bursts and dark energy–dark matter interaction
Also at Instituto de Plasmas e Fusa o Nuclear, Instituto Superior Te cnico
Also at Centro de F sica Teo rica e de Part culas, Instituto Superior Te cnico
Departamento de F sica, Instituto Superior Te cnico
Av. Rovisco Pais 1, 1049-001 Lisboa
Dept. de Matema tica, Univ. Luso fona de Humanidades e Tecnologias
Av. Campo Grande, 376, 1749-024 Lisboa
A B S T R A C T In this work, gamma-ray burst (GRB) data are used to place constraints on a putative coupling between dark energy and dark matter. Type Ia supernovae constraints from the Sloan Digital Sky Survey II (SDSS-II) first-year results, the cosmic microwave background radiation shift parameter from Wilkinson Microwave Anisotropy Probe seven year results and the baryon acoustic oscillation peak from the SDSS are also discussed. The prospects for the field are assessed, as more GRB events become available.
1 I N T R O D U C T I O N
The nature of dark energy and dark matter remains an
outstanding open problem in cosmology. In spite of the success of the
cold dark matter (CDM) parametrization, one must consider more
complex models in order to cast some light on the substance of the
dark components of the Universe. In this work, one considers
models with interacting dark energy and dark matter components. There
are several useful tools to probe the phenomenology of these
models, such as cosmic microwave background radiation (CMBR) data
(Komatsu et al. 2010), baryon acoustic oscillation (BAO)
(Moldenhauer & Ishak 2009; Reid et al. 2010), Supernovae (SNe)
data (Kessler et al. 2009) and the deviation from the virial
equilibrium of galaxy clusters (Bertolami, Gil Pedro & Le Delliou
2007, 2009; Abdalla 2009). It has been suggested (Schaefer 2003;
Bertolami & Tavares 2006) that gamma-ray burst (GRB) may be
used to extend the Hubble diagram to high redshifts, greater than
z = 5. At these epochs the Universe was dominated by dark
matter, from which follows that this tool is less sensitive to dark
energy. However, for models where dark energy and matter are
coupled (Amendola et al. 2003; Bertolami et al. 2007) or unified
(Kamenshcik, Moschella & Pasquier 2001; Bento, Bertolami & Sen
2002, 2003; Barreiro, Bertolami & Torres 2008), GRBs might be a
particularly useful tool (Bertolami & Tavares 2006).
In the late 1960s, the Vela array of military satellites detected
flashes of radiation originating in apparently random directions in
space. The observed bursts lasted between tens of millisecond and
thousands of seconds, and were composed of soft (0.01 to 1 MeV)
gamma-rays. Subsequently space missions such as the US Apollo
programme and the Soviet Venera probes confirmed the existence of
the GRBs, even though its rate of occurrence was virtually unknown
until the deployment of the Compton Gamma-Ray Observatory,
in 1991. This observatory was equipped with a sensitive
gammaray detector, the Burst and Transient Source Explorer (BATSE)
instrument which was able to detect one or two events per day. The
collected data allowed to divide GRBs into two categories: short
duration bursts (short bursts) and long duration bursts (long bursts).
The former usually last for less than two seconds and are dominated
by high-energy photons; the latter last longer than two seconds and
are dominated by lower energy photons. However, this distinction
is not always clear.
The physical origin of GRBs has been debated for a long time,
before their exact position and a reliable estimate of their distance was
lacking (see e.g. Bertolami 1999 and references therein). In 1997,
several GRBs were detected by the BeppoSAX satellite. A GRB
prompt emission is followed by an afterglow emission composed
by all wavelengths. Depending on its brightness, an afterglow can
last from days to months after the burst itself, the transient phase.
The detection of the afterglow did manifold the information on
GRBs. Through their afterglow, GRBs X-ray, optical and radio
counterparts were observed, as well as their redshifts (Fan & Piran
2008), confirming the cosmological origin of most, if not all, GRBs.
When, in 2003, the long GRB 030329 was discovered and linked
with the supernova SN2003dh (Cobb et al. 2004), it became clear
that GRBs are linked with the release of gravitational energy during
the collapse of stellar mass objects. GRBs are most likely
collimated, considering they reach integrated luminosities up to L
1053 erg s 1, making it hard to associate them with an
astrophysical object otherwise. This high energy release creates an outflow
that expands relativistically. Two forms of shocks are ensued by the
burst, the forward shock and the reverse one, which one separated
by a contact discontinuity (Lyutikov 2009). If the ejected plasma
is too strongly magnetized, only the forward shock is formed. One
suggested possibility is that the prompt emission is generated in a
baryon dominated ejecta through internal shocks, while the forward
and reverse shocks yield the long lasting broad-band emission, the
afterglow (Lyutikov 2009). Actually, the full understanding of the
prompt emission mechanism, a basic GRB property, is still lacking.
One possibility is that the prompt emission consists of synchrotron
radiation (Lyutikov 2009), by the relativistic charged particles
moving on the magnetized ejected plasma. Currently, the GLAST/Fermi
mission, in operation, is continuously increasing the available GRB
data and making it worth, as will be discussed and pursued in this
work, considering future prospects for the subject. For an overview
of most recent missions, see e.g. McBreen, Foley & Hanlon (2010)
and references therein.
GRBs can be used as distance indicators (Ghirlanda et al. 2004;
Bertolami & Tavares 2006; Liang & Zhang 2006; Amati et al. 2008).
Its main attractiveness is that the redshift range extends much higher
than that of SNe Ia. The main observables that can be measured
when studying GRBs are its spherical equivalent energy, its peak
isotropic luminosity, the peak energy of its spectrum, the photon
fluence, the energy fluence, the pulse duration and the redshift of
its host galaxy. Several empirical correlations among these
variables can be established. However, there are still large uncertainties
in their calibration. Furthermore, there is still no satisfying
physical mechanism accounting for them, so that assuming that they hold
true can introduce systematic uncertainties in our distance indicator.
From the existing correlations, the very discussed Ghirlanda
relation uses the peak energy of the spectrum, Ep,i and the collimation
corrected energy, E (Ghirlanda et al. 2004). On the other hand, the
Liang and Zang relation correlates the isotropic equivalent energy,
Eiso with Ep,i and the jet break time of the afterglow of the burst
(Liang & Zhang 2006). Finally, the Amati relation, correlates the
isotropic energy, Eiso, with Ep,i (Amati et al. 2008). This relation is
particularly interesting since the Ep,iEiso correlation requires only
two parameters that can be directly inferred from the observations.
This correlation further emphasizes the relevance of the GRB data.
Note that the aforementioned synchrotron process reproduces the
Amati correlation, a quite interesting feature.
In this work, GRB data and the Amati relation, in particular, are
used to probe a generic dark energydark matter interacting model.
In Section 2, the interacting model is presented. The Amati Ep,iEiso
correlation is introduced and discussed in Section 3. The set of real
GRB data is then extended to a mock sample of 500 GRBs using
a method detailed in Section 3.1. In Section 4, we discuss the
constrains obtained from SNe data in Section 4.1, BAO in Section 4.2
and CMBR shift parameter in Section 4.3. In Section 5 we present
the obtained results. In Section 6, conclusions are presented.
2 D A R K E N E R G Y A N D D A R K M AT T E R
I N T E R AC T I O N
The cosmological model consists of homogeneous matter (dark
matter and baryons) and dark energy, where the dark matter and
energy are interacting and have equations of state, pDM = 0 and
pDE = wDE, respectively. The coupled energy densities with a
coupling evolve as follows (Bertolami et al. 2007):
DE + 3H DE(1 + w) = H DM .
The analysis assumes for the ratio of the dark components that
(Bertolami et al. 2007)
for a constant , where a is the scalefactor, assumed that at present
a0 = 1, and z is the redshift.
Inserting the time derivative of equation (3) into equations (1)
and (2), one obtains for the coupling
where 0 = ( + 3w) DE0 . Note that when = 3w, there is no
interaction between dark energy and dark matter, since = 0.
The solutions for equations (1) and (2) can be written as
Inserting equations (3), (5) and (6) into the Friedmann equation for
a flat universe, H2 = H20 (DM + DE + b)/0, where H0 and 0
are the Hubble constant and the total energy density at present, one
E2(z) = (1 + z)3
where b0 and DM0 are the baryon and DM energy densities at
present and E(z) = H(z)/H0.
It is interesting to point out that the Generalized Chaplygin Gas
model (GCG) (Kamenshcik et al. 2001; Bento et al. 2002), an
unified model of dark energy and dark matter, can be seen as a
particular case of this interacting model for = 3(1 + ) and w =
1 (Bento, Bertolami & Sen 2004), being the GCG equation of
state parameter, p = A/, where A is a positive constant.
3 G A M M A - R AY B U R T S
In this work, one considers the Amati Ep,iEiso correlation (for a
discussion of the use of other correlations, see for instance Bertolami
& Tavares 2006 and Liang & Zhang 2006). This correlation can
be used to place constrains on the Hubble diagram. The sample
provided in Amati et al. (2008) and Amati, Frontera & Guidorzi
(2009) that includes the observations of 95 GRB with measurements
for Ep,i, Eiso and redshift, is adopted.
The value of Ep,i is an observable quantity, independent of a
cosmological model. On the other hand, Eiso is computed for each
GRB from its spectral parameters, fluence and redshift using a
specific cosmological model. [The Eiso presented in Amati et al.
(2008, 2009) is computed in the context of the CDM scenario
with h = 0.7, M0 = DM0 + b0 = 0.3 and DE0 = 0.7.]
The Amati correlation assumes a power-law relationship between
Ep,i and Eiso. A cosmological model can then be tested
comparing the Eiso computed from the observations with a theoretical Eiso
obtained from Ep,i. In practice, however, since the Ep,iEiso
relationship is not calibrated, one has to simultaneously fit for the power-law
Furthermore, following Amati (2006), the scatter of the Ep,iEiso
relation that cannot be explained by statistical fluctuations alone
must be taken into account. As in DAgostini (2005) and Amati
(2006), this is done by introducing a third parameter, an extrinsic
variance, ext, in the fitting of the data. Using a power law Ep,iEiso
log Ep,i = m log Eiso + q ,
one aims to minimize the likelihood function, where
and the total variance is 2 = e2xt + p2 + m2 i2so, where p and iso
are the observational variances on log Ep,i and log Eiso, respectively.
The fit is performed for the three parameters m, q and ext.
This minimization is then carried out for different values of the
cosmological parameters, resulting in a profile of the likelihood
3.1 Generating a GRB mock sample
Given that the GRB data are currently rather limited, the analysis
is extended to include a mock sample of GRBs in order to test the
efficiency of the Amati relation on constraining dark energy and dark
matter interacting models. The goal is to check the effect of a larger
number of GRBs, but also the effect of higher redshift GRBs. From
2009 onwards, most of the useful events to fit the Amati relation
came from the Swift and Fermi experiments. One can expect from
these experiments approximately 10 useful GRB events per year.
The future launch of the EXIST mission, scheduled for 2017, will
considerably improve this rate, hopefully simultaneously reducing
the measurement errors. This paper settles on a best case scenario
of 500 GRBs events but conservatively keeping the error bars at the
Following Amati et al. (2008), a distribution mimicking the
observed GRB redshift distribution in the range 0 < z < 6 is used. A
chosen percentage of these events was then replaced with redshifts
uniformly distributed in the range 6 < z < 10. This allows for a
tuning of the number of high-redshift GRBs in the mock sample
that is used in the fits. Note that different choices on the shape of
the high-redshift distribution of GRBs is approximately equivalent
to a change on the redshift cut-off value and its percentage.
Ultimately, only a significant number of high-redshift GRBs can yield
a sizable restriction on 0. The details about the actual low-redshift
distribution used to generate the mock data are discussed in the
Once a redshift distribution is obtained, lognormal distributed
values of Ep,i are attributed to each data point and are associated
with a power law related Eiso. In this paper, a value of 5.86 for
log (Ep,i/1 keV) is used for the mean and a value of 1 is used for the
variance. An extrinsic variance (using ext = 0.41) and Gaussian
errors for Ep,i and Eiso (20 per cent for both) are then included,
mimicking the current observational situation. It is verified that
the results are not particularly sensitive to these choices. Several
mocks with a varying number of low- and high-redshift GRBs were
generated and studied. The presented results consist of a typical
mock sample with 500 events generated and 10 per cent of
highredshift GRBs, in a CDM universe.
4 C O S M O L O G I C A L D ATA
In order to gauge to which extent GRB data can constrain
cosmological parameters, one confronts it with other well-known
cosmologically relevant observational tests such as SNe, BAO and the
CMBR shift parameter.
The SNe sample from the Sloan Digital Sky Survey II (SDSS-II)
first-year results (Kessler et al. 2009) is used, consisting of 288 SNe
Ia with redshifts up to 1.55. SNe are used as distance indicators by
comparing the theoretical distance modulus, th, obtained from the
measured redshift in a given model of cosmological evolution, and
with the inferred distance modulus, obs, computed from fits to the
SNe light curves (using the data of the MLSC2K2 fits found in Kessler
et al. 2009).
Specifically, the theoretical distance modulus is given by
with DL being the scaled (H0 independent) luminosity distance in
DL(z) = (1 + z)
0 E(z )
For each cosmological parameter choice, the used likelihood is
where = fit + disp + z2 is the measurement variance for
2 2 2
obs, including the error from the fit, an intrinsic dispersion error
of disp = 0.16 and a redshift error to account for the host galaxy
peculiar movement and spectroscopic measurement. Only the SNe
results are marginalized over H0 with a flat prior; all the other
constraints assume a constant h = 0.7.
4.2 Baryon acoustic oscillations
One also considers the constraints from the effect of the baryon
acoustic peak of the large-scale correlation function at 100 h1
Mpc separation detected by the SDSS luminous red galaxy sample
(Moldenhauer & Ishak 2009; Reid et al. 2010). The peak position
is related to the quantity
z1 0 E(z)
measured to be A0 = 0.493, with an error of A = 0.017. The used
likelihood is given by
4.3 Cosmic microwave background radiation shift parameter
Here, the constraints from the CMBR Wilkinson Microwave
Anisotropy Probe seven (WMAP7) observations (Komatsu et al.
2010) are considered. The shift parameter (Bond, Efstathiou &
Tegmark 1997) can be used as a distance prior to constrain a given
dark energy model. The shift parameter is given by the equation
with DL being the luminosity distance defined in equation (11), and
z the redshift at decoupling. The standard fitting formula for z
is used (Hu & Sugiyama 1996). This theoretical prediction is then
constrained through the fitted WMAP7 observation, Robs = 1.725
with error R = 0.018, through
5 R E S U LT S A N D C O N S T R A I N T S
One starts with the constraints obtained from the real observed
95 GRBs data (Amati et al. 2008, 2009), combined with the SNe
SDSS-II data (Kessler et al. 2009). For this, M0 = DM0 + b0 is
fixed at M0 = 0.3 with b0 = 0.0445 and the Hubble parameter
held at h = 0.7. The result can be seen in Fig. A1. At 68 per cent
confidence level, the SNe data alone provide the limits 0 [1.93,
1.02] and w [1.07, 0.62]. These results are, by themselves,
highly degenerate in 0. Including the GRB real data, one improves
slightly the constraints to 0 [1.15, 1.66] and w [0.94,
0.58]. As expected, the status of the GRB data at present does
not allow for a significant improvement over the SNe constraints on
these two parameters (see Fig. A1). This SNe degeneracy is
usually lifted combining the SNe data with the CMBR WMAP7 shift
parameter observations yielding the bounds 0 [0.01, 0.13] and
w [0.83, 0.65] at 68 per cent confidence level (not shown).
Even though the present GRB data cannot compete with these
CMBR constraints, with additional data they can provide an
important independent method of lifting the SNe degeneracy in 0.
It must be stressed that these results are obtained for M0 fixed
To illustrate this point, a mock population is chosen in order to
show what can be accomplished from a large population of GRBs,
despite the current level of measurement and theoretical
uncertainties. A CDM universe has been used to generate a mock sample
(see Section 3.1). This time, the results are marginalized over M0
with a flat prior in the interval 0.2 M0 0.4. In Fig. A2, the
results of the SNe and GRB constraints are shown. The constraints
obtained are 0 [0.21, 0.82] and w [0.84, 0.58], at 68 per
cent confidence level. From the SNe data alone, only a lower bound
can be derived for 0 in the considered parameter range.
Once again, this SNe degeneracy in 0 can be lifted by
combining the SNe and CMBR constraints, yielding the bounds 0
[0.29, 0.18] and w [1.08, 0.65] at 68 per cent confidence
level. Fig. A3 combines all the available constraints, namely GRB,
SNe, BAO and CMBR. The CMBR data clearly provides the tighter
constraints on all the parameters. It is, however, highly
degenerate in w and, hence, the SNe constraints on w are required to
yield significant bounds. Notwithstanding, the GRB data have a
similar profile to the CMBR bounds, and they can provide a
significant bound in 0. Thus, GRB data, in combination with SNe,
provide an independent and compatible constraint on 0 and w.
On the other hand, the BAO result clearly does not yield a strong
constraint on the value of either w or 0 (as opposed to the SNe
data), and does not significantly improve the results obtained from
GRBs or CMBR. The overall combined results give the bounds 0
[0.10, 0.08] and w [0.89, 0.70] also at 68 per cent confidence
The GRB constraint on 0 comes mostly from its high-redshift
valued data, whereas the low-redshift valued SNe data present a
degeneracy in 0. Note however that for the GRB and CMBR data,
one encounters a degeneracy in 0 in the form of a bend occurring
at < 0 (i.e. 0 > 3w DE0 ). This occurs as for negative and
for high z, the evolution is dominated by the dark energy density,
rendering the luminosity distance virtually independent of .
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
In this work, the use of GRBs as cosmological tools is considered.
It has been shown (Bertolami & Tavares 2006; Amati et al. 2008)
that GRBs have a great potential to measure the value of DM
independently from the CMBR constraints. In the current work, it
is shown that the present GRB data already give a better constraint
on the dark energy and dark matter coupling parameter 0 than
the BAO results. The SNe results provide good bounds on w, but
are more degenerate in 0. Despite of that, the experimental GRB
sample is still too small to provide significant constraints on 0.
The combined result for SNe and real GRB data further illustrates
this, yielding 0 [1.15, 1.66], with a width of 0 = 2.81, as
opposed to the combined SNe and CMBR data limit 0 [0.01,
0.13] ( 0 = 0.14), both at 68 per cent confidence level. These
results are obtained without marginalization, for fixed DM0 = 0.3.
A mock population of GRBs was then generated showing that
as the number of available events increases, GRB data becomes a
more and more valuable tool in constraining the parameter 0. The
GRBs complement the SNe constraint in a similar way the CMBR
does. For a fixed M0 = 0.3, the SNe and mock GRB data yield
0 = 0.82.
For a deeper insight on the future cosmological implications of the
GRB observations, the analysis was extended to a marginalization
over M0 [0.2, 0.4]. The combined SNe and CMBR results are
0 = 0.47 (68 per cent confidence level), while for SNe and the
mock GRB yield 0 = 1.03. Granting that the obtained bounds
are not as accurate as the CMBR ones, they still provide a valuable
independent measurement of these cosmological parameters.
With the same marginalization in M0 , the combined result for
SNe, CMBR and BAO is 0 [0.27, 0.13] (95 per cent confidence
level). The updated observations improve the previous result 0
[0.4, 0.1] (Guo, Ohta & Tsujikawa 2007). Note, however, that
tighter priors are used in this work and that this model is slightly
different, with the inclusion of non-interacting baryonic matter.
It is interesting to compare the present results with the ones
arising from estimates of the departure from the virial equilibrium
of the Abell cluster A586. The bounds for from the Abell Cluster
A586 yield [3.65, 4.00] ( = 0.35), with w = 1 and z =
0.1708 and M0 = 0.28, (Bertolami et al. 2007). Using SNe and
mock GRB data from the present work and fixing M0 = 0.28, one
encounters [0.93, 2.48] ( = 1.55) at 68 per cent confidence
level. If one considers the particular case of the GCG, the Abell
Cluster A586 limits the parameter to [0.21, 0.33] (68 per
cent confidence level). This compares with the combined SNe and
mock GRB results, [0.25, 0.83] (68 per cent confidence level).
One should bear in mind that the GRB results were obtained from
a mock population and are, thus, only indicative of what can be
expected when the number of observed GRBs increases. The limits
for with the current observational SNe and CMBR data yield
In this work, the Amati correlation (Amati et al. 2008) has been
used, given that it requires only two parameters whose
determination can be inferred and increasing the number of useful GRB
events available. Progress in the calibration and on the
theoretical framework of this calibration would be invaluable, reducing
the error margins in the GRB data and considerably improving the
constraints on the parameters.
AC K N OW L E D G M E N T S
The work of PT was supported by Fundac ao para a Ciencia e
Tecnologia (FCT, Portugal) under the grant SFRH/BD/25592/2005.
The authors would like to thank Sergio Colafrancesco, Giulia Stratta
and Craig Markwardt for discussions about GRB data in the
embrionary phase of this work.
dN (P1 P P2) =
Figure A1. Constraints on the values of w and 0 obtained from the SNe
data (dashed line) and its combination with the 95 GRB data (full line). The
68 and 90 per cent confidence levels are depicted. One fixes M0 = 0.3 and
h = 0.7.
Figure A2. Constraints on the values of w and 0 obtained from SNe and
a mock of 500 GRB population. The wider contour in w corresponds to the
GRBs (in green, for online readers) and the wider contour in 0 to the SNe
(in red, for online readers). Both 68 per cent and 95 per cent confidence
levels are shown. The shaded area refers to the combined confidence region.
A marginalization over M0 [0.2, 0.4] has been considered.
Figure A3. The combined constraints on the values of w and 0 obtained
from the SNe (red), BAO (light blue), CMB (blue) and a mock 500 GRB
population (green). The GRB and SNe contours are the same as in Fig. A2.
The CMB contour is the narrower one (in blue) and the BAO contour is in
background (light blue). The dark patch shows the combined region. The
68 and 95 per cent confidence levels are shown. A marginalization over
M0 [0.2, 0.4] has been carried out.
where dV/dz is the comoving volume element, RGRB is the
comoving GRB rate density and (P) is the detector efficiency as a
function of photon flux. The quantity (L) is the normalized GRB
luminosity function and L is a isotropic equivalent burst
luminosity L = 3200k0e0VkeV E S(E) dE, for the energy E and S(E) is the
rest-frame photon luminosity of the source.
The generation process is performed as follows: using a Monte
Carlo generator, a sample of the desired number of GRBs with z
in the range 0 < z < 6 is generated, using equation (A1). Then,
a desired percentage is replaced randomly by events in the range
6 < z < 10, using a flat distribution. The Eiso is randomly attributed
according to a Gaussian distribution and Ep,i is then calculated using
a power law with the parameters calculated using a fit of the real
GRB sample. The errors are added afterwards assuming a Gaussian
This paper has been typeset from a TEX/LATEX file prepared by the author.