A time–luminosity correlation for γ-ray bursts in the X-rays

Abstract

γ-ray bursts (GRBs) have recently attracted much attention as a possible way to extend the Hubble diagram to a very high redshift. However, the large scatter in their intrinsic properties prevents directly using them as a distance indicator so that the hunt is open for a relation involving an observable property to standardize GRBs in the same way as the Phillips law makes it possible to use Type Ia supernovae as standardizable candles. We use here the data on the X-ray decay curve and spectral index of a sample of GRBs observed with the Swift satellite. These data are used as input to a Bayesian statistical analysis looking for a correlation between the X-ray luminosity L_X(T_a) and the time constant T_a of the afterglow curve. We find a linear relation between log [L_X(T_a)] and log [T_a/(1 +z)] with an intrinsic scatter σ_int= 0.33 comparable to previously reported relations. Remarkably, both the slope and the intrinsic scatter are almost independent on the matter density Ω_M and the constant equation of state w of the dark energy component thus suggesting that the circularity problem is alleviated for the L_X–T_a relation.

cosmological parameters, distance scale, gamma-rays: bursts

1 Introduction

The high-fluence values (from 10⁻⁷ to 10⁻⁵ erg cm⁻²) and the enormous isotropic energy emitted (≃10⁵⁰–10⁵⁴ erg) at the peak in a single short pulse make γ-ray bursts (GRBs) the most violent and energetic astrophysical phenomena. Notwithstanding the variety of their different peculiarities, some common features may be identified looking at their light curves. Although GRBs have been traditionally classified as short and long depending on T₉₀ being smaller or larger than 2 s (with T₉₀ the time over which from 5 to 95 per cent of the prompt emission is released), a recent analysis by Donaghy et al. (2006) has shown that this criterion has to be revised. Indeed, the existence of an intermediated class of GRBs has also been studied (Norris & Bonnell 2006; Bernardini et al. 2007). As a result, the long GRBs are now further classified as a normal and low luminosity with the latter ones probably associated with supernovae (Pian et al. 2006; Dainotti et al. 2007).

Notwithstanding this classification, two phases are clearly visible in the GRB light curve, namely the prompt emission, where most of the energy is released in the γ-rays in only tens of seconds, and an afterglow lasting many hours after the initial bursts. Early observations in the X-rays typically started several hours after the prompt emission so that only the late phase of the light curve could be characterized. It was then found that a phenomenological power law, f(t, ν) ∝t^−αν^−β with (α, β) ≃ (−1.4, 0.9), provided a reasonable fit to the observed data (Piro 2001). However, the launch of the Swift satellite, whose aim is also to observe GRBs X-ray (0.2–10 keV) and optical (1700–6500 Å) afterglows starting few seconds after the trigger, revealed a more complex behaviour. The soft X-ray light curves must indeed be divided in two different classes (Chincarini et al. 2005) according to the steep or mild initial decay. Most of the observed GRB afterglows belong to the first group, showing what has been called a canonical behaviour (Nousek et al. 2006) described by a broken power law. After the initial steep decay (with slope 3 ≤α₁≤ 5), the light curve shows a shallow decay (0.5 ≤α₂≤ 1) followed by a somewhat steeper decay (1 ≤α₃≤ 1.5) beyond 2 × 10⁴ s. These power-law segments are separated by two corresponding break times with t_b1≤ 500 s and 10³≤t_b2≤ 10⁴ s. A new systematic study using GRBs observed with XRT reveals a still more complex behaviour with different power-law slopes and break times (O' Brien et al. 2006; Sakamoto et al. 2007). A significant step forward has been represented by the analysis of the X-ray afterglow curves of the full sample of Swift GRBs showing that all of them may be fitted by the same analytical expression (Willingale et al. 2007, hereafter W07).

Finding out a universal feature for GRBs is the first important step towards their use as a distance indicator. To this aim, one has indeed to look for a universal relation linking observable GRBs properties so that their intrinsic luminosity may be estimated from directly measurable quantities. Previous attempts along this road are represented by the E_iso–E peak (Amati et al. 2002), E_γ–E_peak (Ghirlanda, Ghisellini & Lazzati 2004; Ghirlanda, Ghisellini & Firmani 2006), L–E_peak (Schaefer 2003), L–τ_lag (Norris, Marani & Bonnell 2000), L–V (Fenimore & Ramirez-uiz 2000; Riechart et al. 2001) and L–τ_RT. Moreover, three-parameter relations have also been proposed such as, e.g. the E_iso–E_p–t_b (Liang & Zhang 2005) and that proposed by Firmani and collaborators (Firmani et al. 2005, 2006). On the other hand, some attempts have also been made to compare these empirical correlations with the model dependent ones (Nava et al. 2006; Guida et al. 2008). The above quoted two parameters correlations have then been used by Schaefer (2007, hereafter S07) to construct the first reliable GRBs Hubble diagram extending up to z≃ 6 opening the way towards the use of GRBs as cosmological probes (see e.g. Capozziello & Izzo 2008, and references therein).

In this Letter, we present a possible alternative route towards standardizing GRBs as a distance indicator. To this aim, we use the data in W07 to look for a possible correlation between the X-ray luminosity at the break time T_a and the T_a itself. The data used are presented in Section 2, while Section 3 deals with the statistical tools and results. Conclusions are summarized in Section 4.

2 The Data

W07 have examined the X-ray decay curves of all the GRBs measured by the Swift satellite then available. Their analysis shows that all of them may be well fitted by a simple two-components formula, namely

1

where the first term accounts for the prompt γ-ray emission and the initial X-ray decay, while the second one describes the afterglow. Both components are given by the same functional expression

2

where the transition from the exponential to the power-law decay takes place at the point (T_c, F_c) where the two functional sections match in value and gradient. The parameter α_c determines both the time constant of the exponential decay (given by T_c/α_c) and the slope of the following decay, while t_c marks the initial rise and the time of maximum flux occurring at . Denoting with the suffix ‘p’ and ‘a’ quantities for the prompt and afterglow components, equation (2) may be inserted into equation (1) to give an eight-parameters expression that can be fitted to the X-ray decay curve in order to both validate this expression and determine, for each GRB, the corresponding parameters. Such a task has been indeed performed by W07 using all the 107 GRBs detected by both BAT and XRT on Swift up to 2006 August 1. The fit procedure and the detailed analysis of the results are presented in W07, while here we only remind that the usual χ² fitting in the log (flux) versus log (time) provides estimates and uncertainties on the time parameters (log T_p, log T_a) and the products (log F_pT_p, log F_aT_a).

W07 also performed spectral fitting with xspec (Arnaud 1996) to BAT (for the prompt phase) and XRT (for later phases) data to estimate the spectral index during different phases. Due to the limited frequency range, the GRB spectrum may be simply described by a single power law, Φ(E) ∝E^β, with the slope β depending on the time when the spectrum is observed. W07 reported four different values of β, namely β_p (for the prompt phase), β_pd for the prompt decay, β_a for the plateau observed at the time T_a and β_ad for the afterglow at t > T_a. Actually, the data coverage is not sufficient to measure all of them for the full sample so that, for the weakest bursts, only β_p and β_pd are available. Provided β is known, it is possible to estimate the GRB luminosity at a given time t as

3

where D_L(z) is the luminosity distance at the GRB redshift z and F_X(t) is the flux (in erg cm⁻² s⁻¹) at the time t, K-corrected (Bloom, Frail & Sari 2001) as

4

with (E_min, E_max) = (0.3, 10) keV set by the instrument bandpass. Note that equation (3) is the same as equation (8) in S07, the only difference being the integration limits of the integral at the numerator. Actually, while S07 is interested to the bolometric luminosity, we are here concerned with the X-ray one so that we integrate only over this energy range.

Using the data in W07, we compute the X-ray luminosity at the time T_a so that we have to set f(t) =f(T_a) and β=β_a in equations (3) and (4). Actually, rather than using equation (1), we set f(T_a) =f_a(T_a) since the contribution of the prompt component is typically smaller than 5 per cent, much lower than the statistical uncertainty on f_a(T_a). Neglecting f_p(T_a) thus allows to reduce the error on F_X(T_a) without introducing any bias. This latter error is then estimated by simply propagating those on β_a, log T_a and log F_aT_a thus implicitly assuming that their covariance is null.¹ Should this not be the case, we are underestimating the final error on L_X(T_a). We have, however, checked that our main results are unaffected by a reasonable increase of the errors.

As a final important remark, we note that the presence of the luminosity distance D_L(z) = (c/H₀) d_L(z) in equation (3) constrains us to adopt a cosmological model to compute L_X(T_a). We use a flat Λ cold dark matter (ΛCDM) model so that the Hubble free luminosity distance reads:

5

In agreement with the Wilkinson Microwave Anisotropy Probe (WMAP) 5-yr results (Dunkley et al. 2008), we set (Ω_M, h) = (0.291, 0.697) with h the Hubble constant H₀ in units of 100 km s⁻¹ Mpc⁻¹.

3 A Luminosity–Time Correlation

In order to standardize GRBs to use them as a possible distance indicator, we need to find a correlation between the luminosity and a directly observable quantity. Should such a relation be found, one can then use the observed flux and the estimated L_X to infer D_L(z) and then construct the GRB Hubble diagram. Let us suppose that a power-law relation exists between two quantities R and Q as R=A Q^B. In logarithmic units, this reads log R=a+b log Q with a= log A and b=B. Typically, both R and Q will be known with measurement errors (σ_R, σ_Q) so that the statistical uncertainties on (log R, log Q) will be given by (σ_R/R, σ_Q/Q) (1/ln 10), respectively. These errors may be comparable so that it is not possible to decide what is the independent variable to be used in the usual χ²-fitting analysis. Moreover, the relation R=A Q^B may be affected by an intrinsic scatter σ_int of unknown nature that has to be taken into account. In order to determine the parameters (a, b, σ_int), we can then follow a Bayesian approach (D' Agostini 2005) thus maximizing the likelihood function with

6

with (x_i, y_i) = (log Q_i, log R_i) and the sum is over the objects in the sample. Note that, actually, this maximization is performed in the two-parameter space (b, σ_int) since a may be estimated analytically as

7

so that we will not consider it anymore as a fit parameter.

We use this general recipe to look for a correlation between the X-ray luminosity (in erg s⁻¹) at the time T_a and T_a (in s) itself, i.e. we set y= log [L_X(T_a)] and x= log [T_a/(1 +z)], where we divide time by (1 +z) to account for the cosmological time dilation. Note that, since β_a is needed to compute L_X(T_a), we have to reject most of the 107 GRBs reported in W07 because this is not known. We thus end up with a sample contanining with both log [L_X(T_a)] and log [T_a/(1 +z)] measured.² The Spearman rank correlation turns out to be r=− 0.74 suggesting that a power-law relation between L_X(T_a) and T_a/(1 +z) indeed exists thus motivating further analysis.

We then apply the maximum likelihood estimator described above in order to determine both the slope and the intrinsic scatter of the L_X–T_a correlation thus finding out

Defining the best-fitting residuals as δ=y_obs−y_fit, we can qualitatively estimate the goodness of the fit by considering the median and rms which turn out to be 〈δ〉=−0.08 and δ_rms= 0.52 indeed quite small if compared to the typical log [L_X(T_a)] values. It is also worth noting that δ does not correlate with the other parameters of the fit flux, while the value r=−0.23 between δ and z favours no significative evolution of the L_X–T_a relation with the redshift. The best-fitting relation is superimposed to the data in Fig. 1 where we also present the best fit obtained by the usual χ²-fitting technique. In this case, the best-fitting parameters are obtained by minimizing (through a Levemberg–Marquardt algorithm with 1.5 σ outliers rejection) a χ² merit function given by the second term in equation (6) with , i.e. we (erroneously) assume that there is no scatter and that the errors on log [L_X(T_a)] are negligible. This alternative method gives as best-fitting parameters

in good agreement with the above maximum likelihood estimator so that we argue that our results are independent on the fitting method. However, since the Bayesian approach is better motivated and also allows for an intrinsic scatter, we hereafter elige this as our preferred technique.

Figure 1

Best-fitting curves superimposed to the data with the solid and dashed lines referring to the results obtained with the Bayesian and Levemberg–Marquardt estimator, respectively.

In an attempt to reduce the intrinsic scatter in the above correlation, we have analysed the best-fitting residuals noting that the higher ones are obtained for GRBs with luminosities smaller than 10⁴⁵ erg and time parameter log [T_a/(1 +z)] > 5. We therefore repeated the above analysis using only 28 out of 32 GRBs³ satisfying the two selection criteria 1 ≤ log [T_a/(1 +z)]≤ 5 and [L_X(T_a)]≥ 45. Using the maximum likelihood estimator, we get

with 〈δ〉=− 0.06 and δ_rms= 0.43. The reduced intrinsic scatter and the smaller fit residuals suggest us that, whatever is the unknown mechanism originating the L_X–T_a relation, this is better effective for the class of GRBs satisfying the above selection criteria. The data and the best-fitting curve are shown in Fig. 2 where the dashed line refers to the results obtained with the χ² minimization giving (a, b) = (48.07, − 0.60) reported here for completeness.

Figure 2

Same as Fig. 1, but only using GRBs with 1 ≤ log [T_a/(1 +z)]≤ 5 and log [L_X(T_a)]≥ 45.

The Bayesian approach used here also allows us to quantify the uncertainties on the fit parameters. To this aim, for a given parameter p_i, we first compute the marginalized likelihood by integrating over the other parameter. The median value for the parameter p_i is then found by solving

8

The 68 per cent (95 per cent) confidence range (p_i,l, p_i,h) is then found by solving

9

10

with ɛ= 0.68 (0.95) for the 68 per cent (95 per cent) range, respectively. For the fit to the full data set, we get

while it is

for the selected subsample.

4 Discussion And Conclusion

The high Spearman correlation coefficient, the low value of the fit residuals and the modest intrinsic scatter render the L_X–T_a relation presented above a new valid tool to standardize GRBs. It is worth stressing that L_X–T_a needs only two parameters and one of them is directly inferred from the observations minimizing the effects of the systematic errors. Furthermore, the redshift range covered is large extending from 0.54(0.125) up to 6.6 for the selected (full) sample far beyond the maximum redshift affordable with Type Ia supernovae (SNe Ia) (z≈ 1.7). Should this correlation be confirmed by future higher quality data, one could then combine it with the other relations yet available in literature to work out a GRBs Hubble diagram deep into the matter dominated era thus representing an outstanding cosmological test.

To this end, it is worth comparing the L_X–T_a relation with other ones quoted in literature. When performing such a comparison, however, one should take into account the differences in the cosmological model adopted and the fitting method used. In particular, the choice of how the best-fitting parameters are estimated may have an important impact on the estimate of the intrinsic scatter with the usual χ² fitting leading to an underestimate of σ_int. On the other hand, changing Ω_M in the framework of the flat ΛCDM scenario have a profound impact on σ_int with higher Ω_M giving rise to lower σ_int values (Basilakos & Perivolaropoulos 2008). In order to account for both these issues, one should therefore test all the above correlations using the same statistical tools and cosmological model, a task we will address elsewhere.

As is well known, the paucity of local (i.e. z≤ 0.1) GRBs represents a serious problem for any attempt to standardize GRBs since it is very difficult to directly calibrate any relation. This problem may be partly overcome by fitting the correlation in a subsample of GRBs lying at similar redshift (Amati 2008). However, as a general rule, in order to evaluate the GRB luminosity, a cosmological model has to be adopted thus leading to the circularity problem. Although addressing this problem in detail will be the subject of a forthcoming work, we have here investigated what is the effect of changing the cosmological model by using our maximum likelihood estimator to determine the parameters (a, b, σ_int) as a function of Ω_M in a flat ΛCDM model.⁴ We find the remarkable result that both the best-fitting parameters (b, σ_int) and the rms residual δ_rms are almost insensitive to the value of Ω_M. Indeed, b runs from b≃− 0.590 to −0.565, while σ_int increases from σ_int≃ 0.335 to 0.340 for Ω_M going from 0.2 to 1.0. As a further test, we generalize the ΛCDM model varying not only the matter density parameter Ω_M, but also the equation of state w of the dark energy component (with w=− 1 for the ΛCDM model). For −1.3 ≤w≤− 0.7, neither b nor σ_int significantly changes confirming the qualitative results obtained for the ΛCDM scenario. Although a more detailed analysis is needed, we therefore argue that the circularity problem is alleviated by the use of our L_X–T_a relation.

The encouraging results discussed above are serious arguments in favour of the L_X–T_a relation as a further tool towards the standardization of GRBs as a distance indicator. Should these first evidences be furtherly enforced by more data, the combined use of full set of GRBs correlations discovered insofar could opened the road towards making GRBs the high-edshift analogue of SNe Ia as cosmological probes in the not too distant future.

Acknowledgments

We warmly thank R. Willingale for help with the data and prompt answers to our questions and an anonymous referee for his/her valuable comments.

1

Note that the covariance matrix is not reported in W07, where the parameters of interest are given with their 90 per cent confidence ranges. Following Willingale (private communication), we have assumed independent Gaussian errors and obtained 1σ uncertainties by roughly dividing by 1.65 the 90 per cent errors. Moreover, we preliminary correct for asymmetric errors on log F_aT_a and log T_a (when present) following the prescriptions in D' Agostini (2004).

2

ASCII tables with all the quantities needed for the analysis and the MATHEMATICA codes used are available on request.

3

The four GRBs excluded are GRB050824, GRB060115, GRB060607A and GRB060614. While the first two appear to be unaffected by any problem, for the latter two, the data cover less than 50 per cent of the T₉₀ range. Moreover, for GRB060607A, the prompt component dominates over the afterglowone so that our approximation f (T_a)⋍f_a(T_a) is not valid anymore.

4

To be precise, we let Ω_M run from 0.2 to 1 and adjust h so that Ω_Mh² is fixed to the same value adopted above.