The Three-parameter Correlations About the Optical Plateaus of Gamma-Ray Bursts

Gamma-ray bursts (GRBs) are the most luminous electromagnetic explosive events in the universe. The widely accepted model of this phenomenon is the fireball model, which depicts the erratic, transient events in gamma-rays (Piran 2004; Mészáros 2006; Zhang 2007; Kumar & Zhang 2015), followed by long-lived, decaying afterglows in longer wavelengths (Rees & Meszaros 1992; Mészáros & Rees 1997; Panaitescu et al. 1998; Sari et al. 1998; Zou et al. 2005; Gao et al. 2013; Yi et al. 2013). Lots of afterglow emissions were detected after decades of observations, and have significantly improved our understanding of the physical origin of GRBs. In particular, after the successful launch of the Swift satellite in 2004 (Gehrels et al. 2004), the canonical X-ray light curves were proposed, such as several power-law segments followed by erratic flares (Nousek et al. 2006; Zhang et al. 2006). Actually, the observed optical afterglow is also a mix of various emission components, including the optical flares, the shallow decay segment, the afterglow onset bump, and the late rebrightening component. The physical implications of the flares and the plateau phase in the X-ray and optical bands are discussed often; both phenomena are related to the central engine of the GRB itself (Dai & Lu 1998a, 1998b; Zhang & Mészáros 2001; Burrows et al. 2005; Fan & Wei 2005; Dai et al. 2006; Falcone et al. 2006; Dall'Osso et al. 2011; Rowlinson et al. 2013, 2014, 2017; Wang & Dai 2013; Rea et al. 2015; Yi et al. 2016, 2017b).

The observed GRBs, whose redshifts of z = 8 (Salvaterra et al. 2009; Cucchiara et al. 2011), place GRBs among the farthest known astrophysical sources, indicating GRBs may be good candidates to probe our Universe (Dai et al. 2004). Several interesting empirical correlations have been proposed by the observed GRB data. These physical correlations not only could help the with interpretation of the physical mechanisms responsible for the GRBs, but also can infer important information about the nature of the emitting source (e.g., Wang et al. 2015a for a recent review). Several tight relations were proposed years ago, such as E_γ,iso–E_p,i (also called the Amati Relation, Amati et al. 2002), E_γ–E_p,i (the Ghirlanda Relation, Ghirlanda et al. 2004), and L_iso–E_p,i (the Yonetoku Relation, Yonetoku et al. 2004). Also, some tight correlations about the initial Lorentz factor Γ₀ among E_γ,iso, L_γ,iso and E_p,i were obtained (Liang et al. 2010, 2015; Ghirlanda et al. 2012; Lü et al. 2012; Tang et al. 2015; Yi et al. 2015; Zou et al. 2015). Also, some multi-variable correlations have been found for GRBs (e.g., Liang & Zhang 2005; Rossi et al. 2008), which are useful for understanding GRB physics. A shallow decay (plateau) segment is commonly seen in the X-ray afterglow light curves. Interestingly, a tight correlation has been reported to exist between the break time of the plateau phase (T_b,z, measured in the rest frame) and the corresponding X-ray luminosity (L_b,z, measured in the rest frame) in the X-ray afterglows (the two-dimensional Dainotti relation, Dainotti et al. 2008, 2010, 2011, 2013, 2015, 2017b). Li et al. (2012) also selected a group of optical afterglows with a shallow decay feature, an overplotted L_b,z–T_b,z correlation in the burst frame, and found that optical data share a similar relation to the X-ray data. Later, Xu & Huang (2012) compiled a group of X-ray plateau samples, tried to add a third parameter, i.e., the isotropic energy release E_γ,iso, into the L_b,z–T_b,z correlation, and found that the new three-parameter correlation is much tighter than the previous correlation. Another three-parameter relation, the L_peak–L_b,z–T_b,z relation, is proposed in Dainotti et al. (2016) and Dainotti et al. (2017a), where the L_peak is the peak luminosity in the prompt emission. The tighter new three-parameter correlations may hopefully give a better measure for our universe. For a complete review of GRB correlations, also see Dainotti & Del Vecchio (2017c) and Dainotti et al. (2018).

Therefore, it is interesting to continue to search for possible multi-variable correlations using the optical plateaus, which are useful for understanding GRB physics, and discuss their physical implications for both X-ray and optical plateaus. In this paper, we try to compile a group of the optical afterglows with plateau features, and obtain the results of the optical plateaus by the empirical fitting (Section 2). In Section 3, we present the distributions of GRB optical plateaus, and study the correlations between parameters of the optical plateaus, including two tight, three-parameter correlations about the optical plateaus. Conclusions and a discussion are given in Section 4. A concordance cosmology with parameters ${H}_{0}=71\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$, Ω_M = 0.30, and Ω_Λ = 0.70 is adopted in all parts of this work.

According to Swift observations, lots of GRBs appear at a plateau phase in the early X-ray afterglow, followed by the normal decay phase (or a sharp decay; Nousek et al. 2006; Zhang et al. 2006). A similar shallow decay phase also appeared in the optical light curves, but only a small fraction of optical afterglows has the plateau phase, compared with X-ray light curves (Li et al. 2012). These particular shallow decay phases may have similar physical origins, as both of them are related to the central engine of the GRB itself. The plateau phase is currently understood as being due to ongoing energy injection from the central engine. One reasonable scenario is a fast rotating pulsar/magnetar as the central engine, which spins down through magnetic dipole radiation (Dai & Lu 1998a, 1998b; Zhang & Mészáros 2001; Fan & Xu 2006; Liang et al. 2007; Rowlinson et al. 2013; Lü & Zhang 2014).

In this paper, we try to study the correlations about the optical plateaus by extensively searching for the shallow decay phase in published papers. Well-sampled light curves are available for 50 GRBs that have such a shallow decay segment. Most of the samples are taken from Li et al. (2012; see their Figure 7), and some GRBs are taken from Wang et al. (2015b). According to Dai & Liu (2012), who have investigated this phenomenon, the sufficient angular momentum of the accreted matter is transferred to the newborn millisecond magnetar and spins it up. It is this spin-up that leads to a dramatic increase of the magnetic-dipole-radiation luminosity with time and thus significant brightening of an early afterglow. We also selected some optical light curves with slight rising plateaus at early times. Therefore, the well-sampled afterglows with the obvious plateaus in this paper are the transition in the optical afterglow light curves from a shallow decay (or a slight rising phase) to the normal decay (or an even steeper decay). Some optical light curves are usually composed of one or more power-law segments along with some flares, or rebrightening features, such as GRBs 030723, 081029, 100219A, and so on. Here, we make our fits only around the shallow decay feature, and exclude the mixed components when fitting the light curves. We fit the shallow decay with an empirical smooth broken power-law function (SBPL, Li et al. 2012)

$$\begin{eqnarray}&&{F}_{\mathrm{model}}(t)={F}_{b}{\left[{\left(\displaystyle \frac{t}{{T}_{b}}\right)}^{{\alpha }_{1}\omega }+{\left(\displaystyle \frac{t}{{T}_{b}}\right)}^{{\alpha }_{2}\omega }\right]}^{-\tfrac{1}{\omega }},\end{eqnarray} \tag{ 1 }$$

where α₁ and α₂ are the temporal slopes of the plateau and the followed decay, T_b is the break time, F_b is the optical flux of the break time and ω represents the sharpness of the peak of the light-curve component. Actually, ω = 3 is applied when fitting a light curve. This method is very similar to the fitting method of GRB X-ray plateaus (Dainotti et al. 2016, Dainotti et al. 2017a). We provide the goodness-of-fit test and the residuals in each figures. For the goodness-of-fit test, we take the χ² test:

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{1}^{{{\rm{N}}}_{\mathrm{bin}}}\displaystyle \frac{{[{F}_{\mathrm{obs}}({t}_{i})-{F}_{\mathrm{model}}({t}_{i})]}^{2}}{{[\delta {F}_{\mathrm{obs}}({t}_{i})]}^{2}},\end{eqnarray} \tag{ 2 }$$

where F_obs(t_i) is the observational flux at time t_i, and δF_obs(t_i) is the corresponding error at 68% confidence level. The degrees of freedom (dof) are derived from N_bin − 4, where 4 is the number of free parameters in the SBPL function. We assess a good fit when the value of χ²/dof is close to 1. Next, we calculate the residual as follows:

$$\begin{eqnarray}&&\mathrm{Res}\,=\,\displaystyle \frac{{F}_{\mathrm{obs}}({t}_{i})-{F}_{\mathrm{model}}({t}_{i})}{{F}_{\mathrm{model}}({t}_{i})},\end{eqnarray} \tag{ 3 }$$

which can show the variation of residual flux. Those tests are also plotted in Figure 1.

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View extended figure (5 panels)

The optical plateaus are shown in Figure 1, and the fitting results for the shallow decay segments are summarized in Table 1. We obtain the decay slopes (α₁ and α₂), the break times (T_b), and the corresponding optical flux (F_b) at that moment. The luminosity at the break time (L_b,z) of our sample is derived from

$$\begin{eqnarray}&&{L}_{b,z}=4\pi {D}_{L}^{2}{F}_{b}/(1+z),\end{eqnarray} \tag{ 4 }$$

where z is the redshift, and D_L is the luminosity distance (see also Oates et al. 2009). To produce the luminosity light curves of all optical afterglows, they converted the light curves (in count rate) into flux density and then into luminosity using Equation 2 in Oates et al. (2009). However, we also calculated the luminosity at the optical break time using Equation (4), and the two methods are very similar.

Figure 2 shows the distributions of the break times (T_b), the luminosity at the break time (L_b,z), and the decay slopes (α₁ and α₂). The break time ranges from tens of seconds to 10⁶ s after the GRB trigger, with a typical value of about 10⁴ s, which matches the break time distribution of the X-ray plateaus (Liang et al. 2007; Dainotti et al. 2010; Lü & Zhang 2014). The break luminosity of the optical plateaus mainly ranges from 10⁴⁴ erg s⁻¹ to 10⁴⁷ erg s⁻¹, generally two or three orders of magnitude less than the corresponding luminosity of the X-ray afterglow plateaus. The typical slope values of the two segments are about −0.4 and −1.3, which is consistent with the features of the plateaus.

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Figure 3 presents two correlations about F_b–T_b,z and L_b,z–T_b,z for optical plateaus. L_b,z and T_b,z are all transferred to the rest frame, and the fitting results are shown in Table 2. The break optical flux is anti-correlated with the break of optical plateaus, with a slope index of 0.71. The optical break luminosity L_b,z is anti-correlated with T_b,z, as shown in Figure 3. The best fit shows in Table 2, with a Spearman correlation coefficient of R = 0.84, which is clearly stronger than the F_b–T_b,z correlation. According to Dainotti et al. (2010), who have considered the evolution with the redshift in different bins about the selected sample, they found that the correlation coefficient of the correlation about the luminosity at the break time and break time (hereafter LT) is quite large in the different redshift bins, thus supporting the existence of LT correlation at any redshift. To properly investigate the intrinsic nature of the correlation, it is necessary to apply the Efron & Petrosian (1992) method, which is able to overcome the problem of redshift evolution in variables such as time and luminosity. For an approach successfully tested with this method, see Dainotti et al. (2013) and Dainotti et al. (2015). However, those investigations go beyond the scope of the present paper. The slope of L_b,z–T_b,z is roughly −1, which indicates that the corresponding R-band energy E_R,iso ≡ L_b,zT_b,z is roughly a standard energy reservoir. This tight correlation between L_b,z and T_b,z for optical plateaus is almost the same as the corresponding correlation of X-ray plateaus (Dainotti et al. 2010; Li et al. 2012), and suggests that the longer time of plateau associates with the dimmer break luminosity.

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Interestingly, the subsample of 19 long GRBs associated with supernovae (SNe) also presents a very high correlation coefficient between the luminosity at the end of the plateau and the end time of the plateau of the X-ray afterglows (Dainotti et al. 2017b). Although there is some difference in slope between the normal long GRBs with no SNe and the long GRBs with SNe, and the debate about this difference remains open, it may be resolved with more SNe data, as the tighter LT correlation about long GRBs with SNe is a significant finding and may hopefully give a better measure for our universe.

According to Xu & Huang (2012), they studied a group of X-ray afterglows with plateau features, and added the isotropic energy release into L_b,z–T_b,z. They obtained an even tighter three-parameter correlation called the L_b–T_b–E_γ,iso correlation. In this paper, we try to search for possible multi-variable correlation using the optical plateaus, which depend on the correlation of L_b,z–T_b,z for optical plateaus. Therefore, we selected the isotropic energy for each GRB with the optical plateau (see Figure 1). We investigate whether an intrinsic correlation exists between the three parameters of L_b,z, T_b,z, and E_γ,iso for optical plateaus as

$$\begin{eqnarray}&&{L}_{{\rm{b}},{\rm{z}}}=A+B\mathrm{log}\,{T}_{{\rm{b}},{\rm{z}}}+C\mathrm{log}\,{E}_{\gamma ,\mathrm{iso}},\end{eqnarray} \tag{ 5 }$$

where A, B, and C are constants to be determined from the fit to the observational data. In this equation, A is the constant, while B and C are actually the power-law indices of break time and isotropic equivalent energy when we approximate L_b,z as power-law functions of T_b,z and E_γ,iso. More details can be seen in Xu & Huang (2012) and Liang et al. (2015). In order to find a more significant correlation, we gave the Spearman coefficient and the related hypothesis a test p-value. If the p-value is smaller than 0.1, it means the correlation has a very high probability of being true. At the same time, if the absolute value of the Spearman coefficient is closer to 1, the correlation is tighter. We use an adjusted R² to stand for the goodness of the regression model. An adjusted R² means the variance percentage can be explained by considering the parameter freedom. After that, we performed hypothesis tests for all the regression coefficients and the full linear regression model. Similarly, if the p-value is smaller than 0.1, this means the model has a very high probability of being true. By using the method discussed above, an even tighter correlation about the optical plateau is obtained between the three parameters with

$$\begin{eqnarray}\mathrm{log}{L}_{{\rm{b}},{\rm{z}}} & = & (29.22\pm 5.04)+(-0.92\pm 0.08)\\ & & \times \ \mathrm{log}\,{T}_{{\rm{b}},{\rm{z}}}+(0.37\pm 0.09)\times \mathrm{log}\,{E}_{\gamma ,\mathrm{iso}}.\end{eqnarray} \tag{ 6 }$$

The adjusted R² is 0.77. The F-test p-value for the full linear model is 3.4 × 10⁻¹⁶. The regression coefficient for T_b,z has t-test, the p-value is 6.6 × 10⁻¹⁵. The regression coefficient for E_γ,iso has a t-test where the p-value is 3 × 10⁻⁴. All linear regression models and coefficients pass the hypothesis tests. However, for the L_b,z–T_b,z correlation, the adjusted R² is 0.7. This implies that appending E_γ,iso is meaningful. The fitting result is shown in Figure 4, and it clearly indicates that this three-parameter correlation is tighter than that for L_b,z–T_b,z with the 50 optical plateaus. According to the tight E_γ,iso–E_p,i correlation, called "the Amati Relation" (Amati et al. 2002), we also selected the peak energy (E_p,i) in the ν f_ν spectrum from the literature, and the data are shown in Table 1. Next, we explored the possible correlations among L_b,z, T_b,z, and E_p,i, and found there is also a tight correlation between them, and

$$\begin{eqnarray}\mathrm{log}{L}_{{\rm{b}},{\rm{z}}} & = & (47.48\pm 0.56)+(-0.91\pm 0.09)\\ & & \times \ \mathrm{log}\,{T}_{{\rm{b}},{\rm{z}}}+(0.48\pm 0.16)\times \mathrm{log}\,{E}_{{\rm{p}},{\rm{i}}}.\end{eqnarray} \tag{ 7 }$$

The adjusted R² is 0.75. The F-test p-value for the full linear model is 4.3 × 10⁻¹⁵. The regression coefficient for T_b,z has a t-test p-value of 1.1 × 10⁻¹³. The regression coefficient for E_p,i has a t-test p-value of 4.6 × 10⁻³. All linear regression models and coefficients also pass the hypothesis tests. This correlation also improves when compared with the L_b,z–T_b,z correlation. As shown in Table 2, the Spearman correlation coefficients are 0.89 and 0.87, respectively, with chance probabilities smaller than 10⁻⁴, which suggest these two correlations are quite tight. With the luminosities and energies inside the correlations, these can be used as standard candles. Using these two standard candle relations, one could perform the cosmological parameters by independently comparing with other methods.

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The tight three-parameter correlation for L_b,z, T_b,z, and E_γ,iso is found using X-ray afterglows with a plateau phase (Dainotti et al. 2010; Xu & Huang 2012). In our selected optical sample, the same tight three-parameter correlation exists for the optical plateaus with a Spearman correlation coefficient R = 0.89 (see Table 2). The similar tight correlations between X-ray and optical plateaus indicates that both of them may have the same physical origin. Another tight correlation is obtained among L_b,z, T_b,z, and E_p,i for optical plateaus, with a Spearman correlation coefficient R = 0.87 (see Figure 4 and Table 2). The redshift range covered by our GRB optical sample is not very large, therefore we also consider the evolution with the redshift of the sample. We divided the optical sample into three redshift bins (0–0.98, 1–1.92, 2–4.67) and five redshift bins (0–0.70, 0.72–1.24, 1.25–2.03, 2.1–2.7, 2.71–4.67). The results summarized in Table 3 and shown in Figures 5 and 6 and . The correlation coefficient is quite large in all the redshift bins for both correlations, and the correlations for the subsample are even tighter than those for the full sample. The slopes for the different bins of the two correlations are consistent with the full sample, thus suggesting the existence of the three correlations at any z. We also plotted how the slope of the correlation varies with redshift in different bins in Figure 7.

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The tight correlations of L_b–T_b and L_b–T_b–E_γ,iso are both found in X-ray and optical plateaus, respectively (Dainotti et al. 2008, 2010, 2015, 2017a, 2017b; Li et al. 2012; Xu & Huang 2012). Note that our correlation is about L_b–T_b when considering the prompt isotropic energy E_γ,iso for the optical plateaus, while long GRBs with a plateau phase in their X-ray afterglows also obey the L_b–T_b–L_peak relation (3D relation), where L_peak is the peak luminosity of prompt emission. According to Dainotti et al. (2016) and Dainotti et al. (2017b), the 3D relation planes are not statistically different for sub-categories when the sample is divided into X-ray flashes, GRBs associated with supernovae, ordinary long-duration GRBs, and short GRBs with extended emission. Similar tight correlations between X-ray and optical plateaus indicate that both of them may have the same physical origin. However, this is not the case when the sample of GRBs associated with SNe Ib/c is taken into consideration. In the case of the X-ray correlation the slope is roughly −2 when there is a strong spectroscopic association between GRBs and SNe Ib/c.

Equations (6) and (7) show the coefficients of log T_b,z as −0.92 and −0.91, respectively, which are quite close to −1. We doubt the combination of L_b,z and T_b,z has any correlation with log E_γ,iso and log E_p,i. Then we performed the statistics on the E_R,iso related to log E_γ,iso and log E_p,i respectively. The results are shown in the last two rows of Table 2 and in Figure 8. Some positive correlations can be seen in the figure. The correlations should come from the total kinetic energy. However, the Spearman correlation coefficients are 0.55 and 0.45, respectively, which are clearly looser than the three-parameter correlations. This indicates that the three-parameter correlations are more meaningful. The energy of the R band and the energy of the γ-ray band may not have very tight correlations, which is also true for the peak energy of prompt emission. This might indicate that the prompt emission is not straightforwardly related to the optical emission.

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We have compiled the optical afterglow light curves for 50 GRBs with obvious plateau phases by fitting the light curves with a empirical smooth broken power-law function; we obtain parameters of the optical plateaus such as decay slopes (α₁ and α₂), break times (T_b), and the corresponding optical flux (F_b) at that moment. The break time of optical plateaus ranges from tens of seconds to 10⁶ s, with a typical value of about 10⁴ s, and the corresponding luminosities (L_b,z) mainly range from 10⁴⁴ erg s⁻¹ to 10⁴⁷ erg s⁻¹, generally two or three orders of magnitude less than the corresponding luminosity of the X-ray afterglow plateaus. We added the isotropic energy into the correlation of L_b,z–T_b,z for optical plateaus, and found that the new three-parameter correlation also existed in the GRBs with an obvious optical plateau phase. This L_b,z–T_b,z–E_γ,iso correlation is tighter than the L_b,z–T_b,z correlation for optical plateaus. We next explored the possible correlations among L_b,z, T_b,z, and E_p,i, and found there is also a tight correlation between them. Similar tight correlations between X-ray and optical plateaus indicate that both of them may have the same physical origin. We argue that these two tight L_b,z–T_b,z–E_γ,iso and L_b,z–T_b,z–E_p,i correlations are more physical, which may be directly related to the radiation physics of GRBs. We suggest these two relations can possibly be used as standard candles.

In order to identify shallow decays, one needs to systematically explore temporal breaks in the afterglow light curves. Theoretically, there are another two types of temporal breaks except the shallow decays. The first type is a transition from the normal decay phase (the slope ∼−1) to a steeper phase (the slope ∼−2), which is best interpreted as a jet break (Rhoads 1999; Sari et al. 1999; Frail et al. 2001; Wu et al. 2004; Liang et al. 2008; Racusin et al. 2009; Xi et al. 2017; Yi et al. 2017a). The second type connects the onset afterglow with a smooth bump in the early time. The afterglow onset is produced by a GRB fireball as decelerated by the circumburst medium (Sari & Piran 1999; Molinari et al. 2007; Liang et al. 2010; Yi et al. 2013). Differentiation of these types of breaks is usually straightforward, but sometimes can be more complicated (see Wang et al. 2015b), therefore some disguised breaks with plateau features may also be mixed together with the optical plateaus.

The optical light-curve behavior is quite different from the X-rays. In the X-rays, they are mainly plateaus followed by steep decays, which indicates that the origins before and after the steep decays are different. Including both the X-ray flares and plateaus, they are mainly believed to indicate the long-lasting activity of the central engine, either by direct emission through the late internal shocks, or by electromagnetic energy injection. After the steep decays, they are categorized as normal late afterglow emitted by the external shock into the ambient material. However, from the light curves of the optical band, there is no clear evidence for this gap, even though there are also breaks for many afterglows. Particularly, after the breaks, there are no steep decays followed by normal afterglow decays. These might indicate the different origins for the breaks for X-rays and the optical band. For the X-rays, the breaks mainly show the properties of the central engine, while the optical breaks are mainly a combination of the central engine and the ambient materials. There also exist three-parameter correlations for the optical band, which indicates either the central engine and the ambient for GRBs are connected, or the inner radiation mechanism dominates the correlations. For the former case, the breaks might be caused by the deceleration of the GRB jets, while the deceleration time is contributed by both the total kinetic energy and medium density. For the latter case, the breaks might come from the typical frequencies crossing the observational band. These two hypotheses are distinguishable. One can determine this by checking the spectral indices before and after the breaks. With accumulated data, especially multi-band spectra by more powerful telescopes, one may reveal the inner mechanism. The most promising scenario might be that the optical breaks are divided into several groups by taking care of the spectral index and the light-curve shapes. The correlations of sub-groups might be even tighter. This may give yet another and even tighter standard candle relation.

We thank the anonymous referee for constructive suggestions. This work is supported by the National Basic Research Program of China (973 Program, grant No. 2014CB845800), the National Natural Science Foundation of China (grants No. 11703015, 11547029, 11673006), the China Postdoctoral Science Foundation (grant No. 2017M612233), the Natural Science Foundation of Shandong Province (grant No. ZR2017BA006), and the Programs of Qufu Normal University (xkj201614, 201710446105), the Guangxi Science Foundation (grant No. 2016GXNSFFA380006), the Innovation Team and Outstanding Scholar Program in Guangxi Colleges, and the One-Hundred-Talents Program of Guangxi colleges.