Gamma-ray Bursts Cosmology with The X-ray Fundamental Plane Relation


 Cosmological models and the value of their parameters are at the center of the debate because of the tension between the results obtained by the SNe Ia data and the Plank ones of the Cosmic Microwave Background Radiation. Thus, adding cosmological probes observed at high redshifts, such as Gamma-Ray Bursts (GRBs), is needed. Using GRB correlations between luminosities and a cosmological independent variable is challenging because GRB luminosities vary widely. We corrected a tight correlation between the rest-frame end time of the X-ray plateau, its corresponding X-ray luminosity, and the peak prompt luminosity: the so-called fundamental plane relation, using the jet opening angle. Its intrinsic scatter is 0:017 m 0:010 dex, 95% smaller than the isotropic fundamental plane relation, the smallest compared to any current GRB correlation in the literature. This shows that GRBs can be used as reliable cosmological tools. We use this GRB corrected correlation for the so-called platinum sample (a well-defined set with relatively flat plateaus), together with SNe Ia data, to constrain different cosmological parameters like the matter content of the universe today, M, the Hubble constant H0, and the dark energy parameter w for a wCDM model. We confirm the wCDM model but using GRBs up to z = 5, a redshift range much larger than one of SNe Ia.

tools, we need to understand their emission mechanisms. Indeed, there is still an ongoing debate regarding their physical mechanisms and their progenitors. There are several proposed scenarios regarding their possible origin: e.g., the explosions of extremely massive stars and the merging of two compact objects, like neutron stars (NSs) and black holes (BHs). Both these models consider ordinary NSs, BHs or fast spinning newly born highly magnetized NSs (magnetars) as the central engines of the GRBs' energy emission. In the former scenario, the compact object acting as the central engine is the remnant of the massive star after its collapse, while the latter results from the coalescence of the two compact objects and its subsequent explosion.
To pinpoint the different possible origins, we need to categorize GRBs according to their phenomenology. The GRB prompt emission is usually observed from hard X-rays to ≥ 100 MeV γ-rays, and sometimes also in optical wavelengths. The afterglow is the long-lasting multiwavelength emission (in X-rays, optical, and sometimes radio) following the prompt.
GRBs are traditionally classified as short and long GRBs, depending on the prompt emission duration: T 90 ≤ 2 s or T 90 ≥ 2 s 1 , respectively 2, 3 .
The Neil Gehrels Swift Observatory (hereafter Swift) records the observations of the X-ray plateau emission 4-6 which generally lasts from 10 2 to 10 5 s and is followed by a power law (PL) decay phase. Several models have been proposed to explain the plateau: the long-lasting energy injection into the external shock, where a single relativistic blast wave interacts with the surrounding medium 7 or the spin-down luminosity of a newly born magnetar 8 . Several correlations involving 1 T 90 is the time over which a burst emits from 5% to 95% of its total measured counts in the prompt emission. 3 the plateau [9][10][11][12][13] and their applications as cosmological tools have been discussed in the literature so far [14][15][16] . One remarkable correlation which involves the plateau emission is the so-called Dainotti relation, which links the rest frame time at the end of the plateau emission, T * X , with its correspondent luminosity, L X 10 . This correlation can be explained naturally within the magnetar scenario 8,17 , and it indicates that the energy reservoir of the plateau is constant. An extension of this correlation in three dimensions has been discovered by adding the prompt emission's peak luminosity, L peak 9, 18 . The Dainotti 3D relation, the so-called fundamental plane correlation, defines a plane whose axes are L X , T * X and L peak .
To boost the use of this correlation as a cosmological tool, as pointed out in 9, 10, 18 , we need to select a subsample of GRBs with very well-defined properties from both a morphological and a physical point of view. We focus our attention on a sample with well-defined and almost flat plateaus, called the platinum sample, whose features are detailed in Methods.
However, the mentioned relations and many others do not consider the collimated nature of GRBs, which leads to an overestimation of the luminosities and the energies. The collimated nature of GRBs is widely accepted from a theoretical point of view: their emission is beamed into a jet, enclosed into the so-called jet opening angle (θ jet ) 19 . This fact requires the isotropic luminosities, L iso , to be corrected in the following way: L jet = L iso (1 − cos(θ jet )), which can decrease the inferred luminosities associated with GRBs by two or three orders of magnitude concerning a more simplified isotropic assumption. Thus, we here use the tightest three-parameter correlation in the literature, which is the fundamental plane relation introduced before, and we correct it for θ jet , see Methods . First, we have selected 214 GRB X-ray plateau afterglows detected by Swift from 2005   January up to 2019 August with known redshifts, spectroscopic or photometric, available in 20 , on the Greiner web page 2 and in the Gamma-ray Coordinates Network (GCN) circulars and notices 3 , excluding redshifts for which there is only a lower or an upper limit. We include all GRBs for which the Burst Alert Telescope (BAT) and X-Ray Telescope (XRT) light curves can be fitted by the phenomenological 21 model, see Methods. We then choose from the whole sample a subset of long GRBs with very well-defined selection criteria, the so-called platinum sample. Fig. 1 shows the isotropic fundamental plane correlation for the platinum sample in 3D (upper panel), and the same correlation corrected by θ jet (lower panel). We see a great reduction of the intrinsic scatter, σ int (95%), from the isotropic to the jetted fundamental plane.
The isotropic fundamental plane relation can be written in the following way: where a iso and b iso are the best fit parameters given by the D'Agostini 22 fitting related to log T * X and log L peak , respectively, while C iso is the normalization. The best fits are a iso = −0.87 ± 0.11, b iso = 0.54 ± 0.07, C iso = 22.95 ± 3.85, and σ int = 0.34 ± 0.04.
The fundamental plane relation with the correction introduced by θ jet can be written as follows: where a jet and b jet are the best fit parameters given by the D'Agostini fitting related to log T * X and 2 https://www.mpe.mpg.de/ jcg/grbgen.html 3 http://gcn.gsfc.nasa.gov/ 5 log L peak , respectively, while C jet is the normalization. The best fits are a jet = −0.93 ± 0.03, b jet = 0.25 ± 0.03, C jet = 36.70 ± 1.50, and σ int,jet = 0.017 ± 0.010. The contour plots of these parameters are shown in Fig. 4.  Table 1.
We stress that these results are compatible with the ΛCDM model. The great advantage is that we have computed them with high-z probes up to z = 5. The reduction of σ int after the correction introduced by θ jet allows GRBs to be reliable standard candles, together with SNe Ia, to provide a more precise estimate for these cosmological parameters.

References
SNe Ia sample

The sample Selection
We analyzed the light curves from 2005 January until 2019 August taken from the Swift web page repository, 4 and we derived the spectral parameters following 4 .
We fit this sample with the W07 functional form for f (t) which reads as follows: 4 http://www.swift.ac.uk/burst analyser modeled for both the prompt (index 'i=p') γ -ray and initial X-ray decay and for the afterglow ('i=X'), so that the complete light curve f tot (t) = f p (t) + f X (t) contains two sets of four free After we gather these parameters we proceed with the computation of the peak prompt luminosity at 1 second, L peak , and the luminosity at the end of the plateau emission, L X , using the following equation: where F = F X , F peak , with F X the measured X-ray energy flux at the end of the plateau phase and M pc −1 , T * X is the rest frame time at the end of the plateau and K is the K-correction for the cosmic expansion defined in Eq. 5 shown below 25 . For Swift GRBs, K is calculated in the following way 25 : where Φ(E) is the functional form for the spectrum, for which we assume either a PL for the plateau phase and a cutoff power law (CPL) for the prompt emission. In a few cases when the CPL for the prompt emission is not a viable fit we use the PL.

13
To further create a sample with more homogeneous spectral features, and hence restricting the analysis to a more uniform class of objects, we consider the GRBs for which the spectrum computed at 1 s has a smaller χ 2 for the CPL fit than for a PL. Specifically, following 26 , when the χ 2 CP L − χ 2 P L < 6, either a PL or a CPL can be chosen, since the goodness of the fit is equivalent, and in these cases we here chose the CPL. In addition, for all GRBs for which the CPL was absent or the χ 2 satisfies this criterion, there is no substantial difference in the spectral fitting results if one considers either a PL or a CPL. We have discarded six GRBs that were better fitted with a black body model than with a PL or a CPL.
From a total sample of 214 GRBs presenting plateaus, we choose a subsample by considering strict data quality and the following morphology requirements: • the beginning of the plateau must have at least five data points; • the plateau must not be too steep (the angle of the plateau has to be less than 41 • ); • T X must not be inside a large gap of the observed data; • The plateau duration must be (> 500 s)and the plateau must not contain flares or bumps.
The light curves with these features create a sample of 50 platinum GRBs. It is important to note that the criteria defining our sample are objectively determined before the construction of the correlations sought; sample cuts are introduced strictly following either data quality or physical class constraints. The furthest GRB belonging to the platinum sample is found at z = 5.

The Angle Computation
The problem is that there are only a few observed jet opening angles, θ jet . To overcame this issue we use an indirect method to estimate these quantities built upon the method of Pescalli 2 7, which, in turn, is based on the ratio between the E peak − E iso 2 8 and the E peak − E γ,iso correlations 2 9, where E peak , E iso and E γ,iso = E iso × (1 − cos θ) are the peak of the νF ν spectrum, the isotropic energy and the collimated energy, respectively. All these quantities are computed for the prompt emission. The Pescalli method involves prompt emission variables, and we improve upon this using the fundamental plane relation, which also considers plateau emission variables.
More specifically, we assume, analogously to the Pescalli derivation, that the fundamental plane relation is a general correlation, but the scatter that we infer for it is not only due to the intrinsic physical scatter, at least not for the majority of it, but also due to the fact we are not correcting the luminosities for the jet opening angle.
Thus, we add a correction to the Pescalli estimation of the angle due to the luminosity at the end of the plateau emission. This correction, which we call vertical shift, s, represents how much a jetted emission would influence the determination of the true luminosity.
We calculate, as a first approximation, the values of the jet opening angle using the method described in 27 with the following equation: where C is a normalization constant, which here is defined as the minimum value of L iso = 15 E iso /T 90 for a given sample, and ξ = 0.3, following 27 . We compute E iso as the following: where S is the fluence taken from the Third Swift BAT Gamma-Ray Burst Catalog 3 0 and K is the K-correction already defined above.
To give generality to our method, we have simulated all the parameters involved in our fitting and in Eq. 6. These parameters are derived by the best fit distributions computed from the observed data of the platinum sample.
We use a new approach to compute the jet opening angle refining Eq. 6 by using the vertical shift from the three-dimensional fundamental plane relation. We can leverage from the previous Eqs., for which 1 − cos(θ jet ) ∝ L iso , and from the fact that, observationally, log L iso ∝ β × log L peak . Since the fundamental plane relates log L peak with log L X and log T * X , we can add to the derivation of Pescalli 27 the information given by the fundamental plane correlation in the following way: where α × s is the correction factor that we apply to the Pescalli formula, see Eq. 6, and C and ξ are the new coefficients obtained by fitting the Pescalli angle with the vertical shift and L peak . The vertical shift is the defined as s = log L X − (a sim,platinum × log T * X,sim + b sim,platinum × log L peak,sim + C 0,sim,platinum ) (9)   16 where a sim,platinum , b sim,platinum and C 0,sim,platinum are the best fit parameters of the simulated isotropic fundamental plane relation for the platinum sample (Eq. 1), L peak,sim and T * X,sim are the simulated peak prompt luminosity and end time of the plateau emission of the GRBs belonging to the platinum sample. We stress again that we use simulations to give generality to the definition of the vertical shift. More specifically, to guarantee the generality of this equation, we have simulated the best fit distribution parameters of the observed data of the platinum sample 10000 times.
Then, Eq. 9 is substituted into Eq. 8 to compute the coefficients C , ξ and α. We obtain these coefficients by fitting Eq. 8 and repeat the fitting 10000 times to compute the mean of these best fit parameters as the final values. This procedure will guarantee that the values computed are statistically meaningful over repeated iterations to correctly compute the errors associated with them.
To compute the jet opening angle for each GRB in our sample, we use Eq. 8 to have a one to one correspondence of each GRB with its jet opening angle.
We then substitute (1 − cos θ) in the jetted fundamental plane, see Eq. 2 and, thus, we obtain a new set of best fit parameters with a much smaller intrinsic scatter with respect to the isotropic fitting.
Indeed, from Figs. 3 and 4 , we see that the fundamental plane for the platinum sample corrected for the jet opening angle (its projection in 2D is shown in the lower panel of Fig. 3) leads to a remarkable reduction of σ int compared to the isotropic fundamental plane (its 2D projection is shown in the upper panel of Fig. 3). In fact, we reduce σ int of 95% from 0.34 ± 0.04 to 0.017 ± 0.010. The contour plots of the best fit parameters are shown in Fig. 4.
To find these best fitting parameters we use the D'Agostini 22 Bayesian method, which takes into account the error bars on all the axes and it also includes σ int . Uncertainties are always quoted in 1 σ.
The 3D Relation for platinum GRBs with correction for θ jet as a cosmological tool.
Here we summarize the method used to obtain cosmological parameters by using the 3D fundamental plane with the platinum sample by introducing the θ jet correction together with SNe Ia data.
The values of θ jet are derived from Eq. 8.
where c is the speed of light and z is the redshift of the GRB for which we are computing the distance. We recall that in our Bayesian approach, we let Ω M vary simultaneously with the other variables, so that we do not incur into the so-called circularity problem. This means that In our cosmological computations, we use both the GRBs belonging to the platinum sample and the SNe Ia data of the Pantheon sample 23 . To evaluate the best cosmological parameters underlying our universe, we make use of the distance moduli, µ obs,SN e , derived from the observations of SNe Ia taken directly from 23 . Regarding the µ obs,GRBs of GRBs, we derive it performing algebraic manipulations on the fundamental plane relation corrected for the jet opening angle that lead to the following formula: where log F p,cor and log F a,cor are the fluxes in the prompt and afterglow emission corrected for the K-correction and the jet opening angle. We then compare both µ obs,GRBs,SN e with the theoretical µ th , defined as follows (with the appropriate dimensional units): We have combined the two likelihoods for our Bayesian approach, one related to the GRBs while the other to the SNe Ia. The total likelihood is: where the first term is the likelihood related to GRBs' distance moduli 16,32 , and the second refers to the likelihood of SNe Ia data, where C inv is the inverse of the covariance matrix of the SNe data 23 .
We   (1) we obtain a 11% decrease in the scatter for Ω M .
The crucial point of our results is that we are able to obtain compatible cosmological parameters with better precision in some cases by also using GRBs and with the great advantage of using cosmological probes which are observed at very high redshift, in this case of our sample up to redshift 5.