A Large Catalog of Multiwavelength GRB Afterglows. I. Color Evolution and Its Physical Implication

The extinction due to dust from the host galaxy can significantly affect the observations. For wavelength λ_i, the dust extinction is defined as

$$\begin{eqnarray}&&A({\lambda }_{i})={m}_{{\rm{O}}}({\lambda }_{i})-{m}_{{\rm{T}}}({\lambda }_{i}).\end{eqnarray} \tag{ 1 }$$

The observed SED of the optical afterglow is modeled by an absorbed power law in frequency (Kann et al. 2006)

$$\begin{eqnarray}&&{F}_{\nu }={F}_{0}{\nu }^{-{\beta }_{o}}{e}^{-\tau ({\nu }_{\mathrm{host}})},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&\tau ({\nu }_{\mathrm{host}})=\displaystyle \frac{1}{1.086}{A}_{{\rm{V}}}\eta ({\nu }_{\mathrm{host}}),\end{eqnarray} \tag{ 3 }$$

in which β_o is the intrinsic power-law slope of the SED, F₀ is the normalization constant, $\eta ({\nu }_{\mathrm{host}})={A}_{\lambda }^{\mathrm{host}}/{A}_{{\rm{V}}}^{\mathrm{host}}$ is given by the extinction model assumed for the GRB host galaxy, and ${A}_{\lambda }^{\mathrm{host}}$ is the host galaxy extinction correction (known only with large uncertainties). We do not carry out spectral fitting in this paper. Instead, the host galaxy extinction ${A}_{{\rm{V}}}^{\mathrm{host}}$, the spectral index β_o, and the redshift z are acquired from the literature, and their values are given in Table 1.

Table 1.  Properties of the GRB Sample with Multicolor Light Curves

GRB	Filters	${E}_{B-V}^{G}$	βo	${A}_{{\rm{v}}}^{\mathrm{host}}$	z	References
		(Galaxy)	(Spectral Index)	(Host Galaxy)	(Redshift)	(for Data Sources)
970508	UBVRcIc	0.034	1.11	0.38 ± 0.11	0.8349	1, 2, 3, 4, 5, 6
980425	UBVRcIc	0.059	⋯	0.17 ± 0.02	0.0085	7, 8, 9, 10
990510	BVRIJHKs	0.226	0.55	0.22 ± 0.07	1.6187	11, 12, 13, 14
990712	VRI	0.035	0.99 ± 0.02	0	0.4331	15, 16, 17
000301C	UBVRIJK	0.053	0.70	0.12 ± 0.06	2.0404	18, 19, 20, 21, 22, 23
000926	UBVRI	0.025	1.00 ± 0.18	0.15 ± 0.07	2.0387	24, 25, 26
021004	UBVRIJHK	0.074	0.14	0.05	2.3304	27, 28, 29, 30, 31
030226	UBVRIJHK	0.023	0.70 ± 0.03	0.06 ± 0.06	1.98691	32, 33, 34
030328	UBVRI	0.075	0.36 ± 0.45	0.05 ± 0.15	1.5216	35
030329	UBVRIJH	0.029	0.54 ± 0.02	0.39 ± 0.15	0.16867	36, 37, 38, 39
050525A	UVW2UVM2UVW1UBVRIJH	0.117	0.97 ± 0.10	0.36 ± 0.05	0.606	40, 41, 42, 43, 44
050801	UVW2UVM2UVW1UBVRIcJK	0.081	1.00 ± 0.16	0.30 ± 0.18	1.56	43, 45, 46
050820A	UBVRIJHKg'z'	0.054	0.72 ± 0.03	0.07 ± 0.01	2.61469	43, 47
050922C	UBVRI	0.093	0.51 ± 0.05	0.01 ± 0.01	2.198	43, 48
051111	BVRI	0.146	0.76 ± 0.07	0.20 ± 0.10	1.54948	49, 50, 51, 52
060218	UVW2UVM2UVW1UBVRIJHK	0.124	⋯	0.13 ± 0.01	0.03342	53, 54, 55, 56
060418	BVRIJHKz'	0.319	0.78 ± 0.09	0.20 ± 0.06	1.49000	43, 57, 58
060526	whiteBVRIJHKs	0.078	0.51 ± 0.32	0.05 ± 0.11	3.221	59, 60
060607A	whiteUBVgriH	0.027	0.72 ± 0.27	0.08 ± 0.08	3.0749	58, 61, 62, 63
060614	whiteUVW2UVM2UVW1UBVRIJK	0.017	0.47 ± 0.04	0.28 ± 0.07	0.1254	64, 65, 66, 67, 68
060729	whiteUVW2UVM2UVW1UBVCR	0.05	0.78 ± 0.03	0	0.5428	69, 70
060906	g'Rci'z'	0.087	0.56 ± 0.02	<0.09	3.6856	71
060908	whiteUBVRIJHKgriz	0.027	0.30	0.05 ± 0.03	1.8836	72
061007	BVRi	0.019	0.78 ± 0.02	0.48 ± 0.10	1.2622	73
061126	UVM2UVW1UBVRIJKHgri	0.187	0.95	0.10 ± 0.06	1.1588	74, 75
070125	UVW2UVM2UVW1UBVRIJHKsgriz	0.052	0.55 ± 0.04	0.11 ± 0.04	1.54705	76, 77
071010A	clearUBVRIJHK	0.114	0.68	0.64 ± 0.09	0.985	78
071025	JHKs	0.059	0.42 ± 0.08	0.12 ± 20.6	5.2	79
071031	g'r'i'z'JHKs	0.008	0.64 ± 0.01	0.14 ± 0.13	2.692	80
080109	uvw2uvm2uvw1UBVRIJHKri	0.02	⋯	⋯	0.007	81
080310	clearBVRIJHKwhiteubv	0.043	0.42 ± 0.12	0.19 ± 0.05	2.42743	43, 82, 83
080319B	whiteclearUVW2UVM2UVW1UBVRIJHKsgriz	0.011	0.66	0.06 ± 0.03	0.9371	84, 85, 86
080330	UVW1UBVRIJHKgrizwhiteclear	0.016	0.49	0.19 ± 0.08	1.5115	43, 87
080413B	g'r'i'z'	0.032	0.25 ± 0.07	0	1.1014	88
080603A	clearg'r'i'BVRIJHK	0.044	0.98 ± 0.04	0.90 ± 0.19	1.67842	89
080607	g'r'i'z'clearVRIJHKs	0.023	1.32 ± 0.07	3.3 ± 0.4	3.306	90
080710	g'r'i'z'JHKs	0.046	0.80 ± 0.09	0.11 ± 0.04	0.845	91
080810	whiteBVRIunfitbvir	0.031	0.44 ± 0	0.16 ± 0.02	3.35104	43, 92
081007	g'r'i'z'JHK	0.015	0.47 ± 0.09	0.31 ± 0.25	0.5295	93
081008	UVW1UBVwhiteRcCRIcJHKg'r'i'z'	0.095	1.10 ± 0.04	0.17 ± 0.11	1.9683	43, 94
081029	g'r'i'z'JHKs	0.028	1.00 ± 0.01	0.03 ± 0.02	3.8479	95
090102	g'i'z'RJHKs	0.045	0.74	0.12 ± 0.11	1.548	96
090426	g'r'i'z'R	0.015	0.76 ± 0.14	0	2.609	97, 98, 99
090510	g'r'i'z'	0.02	0.85 ± 0.05	0.17 ± 0.21	0.903	100
090618	uvw2uvm2uvw1uBVwhiteRc	0.085	0.5	0.3 ± 0.1	0.54	101, 102
090926A	g'r'i'z'	0.023	0.72 ± 0.17	0.13 ± 0.06	2.1071	103, 104
091018	g'r'i'z'JHKs	0.034	0.57 ± 0.01	0.09	0.9710	105
091024	BVRI	0.98	⋯	⋯	1.0924	106
091029	g'r'i'z'	0.013	0.49 ± 0.12	0	2.752	107
091127	g'r'i'z'	0.037	0.43 ± 0.10	0.17 ± 0.15	0.49044	108
100316D	BVRIg'r'i'z'JH	0.117	0.94 ± 0.05	0.43 ± 0.03	0.0592	109
100621A	g'r'i'z'JHKs	0.028	0.80 ± 0.10	3.65 ± 0.06	0.542	110
100814A	g'r'i'z'JHKs	0.02	0.47 ± 0.05	<0.04	1.44	111
101219B	g'r'i'z'JHK	0.02	0.62 ± 0.01	<0.1	0.55185	112
110205A	whiteubvgrizRJHK	0.098	⋯	0.35 ± 0.03	2.21442	113
110213A	whiteclearubvgrizBVRI	0.432	1.22 ± 0.18	⋯	1.4607	110, 114, 115, 116, 117
110918A	g'r'i'z'JHKs	0.015	0.70 ± 0.02	0.16 ± 0.06	0.984	118
111209A	UVW2UVM2UVW1UBVWhiteg'r'i'z'JHK	0.02	1.07 ± 0.15	0.3-1.5	0.677	119, 120, 121
120119A	clearBVRIJHK	0.109	0.92 ± 0.02	0.62 ± 0.06	1.728	122
120404A	g'r'i'z'UBVRIJH	0.052	1.04 ± 0.02	0.07 ± 0.02	2.8767	123
120422A	g'r'i'BVRI	0.068	0.50	0	0.28253	124, 125
120711A	g'r'i'z'JHKs	0.075	0.53 ± 0.02	0.85 ± 0.06	1.405	126
120729A	g'r'i'z'VR	0.162	1.00 ± 0.10	0.15	0.80	127
120815A	g'r'i'z'	0.096	0.78 ± 0.01	0.15 ± 0.02	2.3586	128
121024A	g'r'i'z'JHKs	0.128	0.86 ± 0.01	0.18 ± 0.04	2.3010	129
121217A	g'r'i'z'	0.392	0.87 ± 0.04	0 ± 0.03	3.08	130
130215A	g'r'i'z'JHK	0.137	0.90 ± 0.20	0	0.597	127
130427A	g'r'i'z'	0.022	0.92 ± 0.10	0.13 ± 0.06	0.3399	131, 132
130702A	clearu'g'r'i'z'UBVRJH	0.031	0.52 ± 0.19	0.30 ± 0.07	⋯	133, 134, 135
130831A	g'r'i'z'BRcIc	0.054	0.85 ± 0.01	0.06 ± 0.04	0.4791	123, 136, 137
130925A	g'r'i'z'JHK	0.018	0.32 ± 0.03	5.00 ± 0.70	0.3479	138

References. References for z: GRB 970508: Bloom et al. (1998); GRB 980425: Galama et al. (1998); GRB 990510, GRB 990712: all from Vreeswijk et al. (2001); GRB 000301C: Jensen et al. (2001); GRB 000926: Castro et al. (2003); GRB 021004: Castro-Tirado et al. (2010); GRB 030226: Shin et al. (2006); GRB 030228: Maiorano et al. (2006); GRB 030329: Thöne et al. (2007); GRB 050525A: Della Valle et al. (2006b); GRB 050801: de Pasquale et al. (2007); GRB 050820A, GRB 060607A, GRB 060729, GRB 060906, GRB 060908, GRB 061007, GRB 071025, GRB 071031, GRB 080413B, GRB 080710: all from Fynbo et al. (2009); GRB 050922C: Piranomonte et al. (2008); GRB 051111: Penprase et al. (2006); GRB 060218: Pian et al. (2006); GRB 060418: Vreeswijk et al. (2007); GRB 060526: Jakobsson et al. (2006); GRB 060614: Della Valle et al. (2006a); GRB 061126: Perley et al. (2008); GRB 070125: De Cia et al. (2011); GRB 071010A: Covino et al. (2008); GRB 080109: Modjaz et al. (2009); GRB 080310: De Cia et al. (2012); GRB 080319B: D'Elia et al. (2009b); GRB 080330: D'Elia et al. (2009a); GRB 080603A: Guidorzi et al. (2011); GRB 080607: Prochaska et al. (2008); GRB 080810: Page et al. (2009); GRB 081007: Berger et al. (2008); GRB 081008: D'Avanzo et al. (2008); GRB 081029: Holland et al. (2012); GRB 090102: de Ugarte Postigo et al. (2012); GRB 090426: Levesque et al. (2010); GRB 090510: McBreen et al. (2010); GRB 090618: Cano et al. (2011b); GRB 090926A: D'Elia et al. (2010); GRB 091018: Wiersema et al. (2012); GRB 091024, GRB 091029: all from Virgili et al. (2013); GRB 091127: Vergani et al. (2011); GRB 100316D: Bufano et al. (2012); GRB 100621A: Milvang-Jensen et al. (2010); GRB 100814A: O'Meara et al. (2010); GRB 101219B: Sparre et al. (2011); GRB 110205A, GRB 110213A: all from Cucchiara et al. (2011); GRB 110918A: Elliott et al. (2013); GRB 111219A: Vreeswijk et al. (2011); GRB 120119A: Cucchiara & Prochaska (2012); GRB 120404A: Guidorzi et al. (2014); GRB 120422A: Schulze et al. (2014); GRB 120711A: Tanvir et al. (2012); GRB 120729A: Tanvir & Ball (2012); GRB 120815A: Krühler et al. (2013); GRB 121024A: Friis et al. (2015); GRB 121217A: Elliott et al. (2014); GRB 130215A: Cucchiara & Fumagalli (2013); GRB 130427A: Flores et al. (2013); GRB 130702A: Kelly et al. (2013); GRB 130831A: Cucchiara & Perley (2013); GRB 130925A: Schady et al. (2015). References for ${A}_{v}^{\mathrm{host}}$: GRB 970508, GRB 000301C, GRB 000926, GBR 021004, GRB 030226, GRB 030328, GRB 030329: all from Kann et al. (2006); GRB 990712: Kann et al. (2016); GRB 980425, GRB 990510, GRB 050525A, GRB 050801, GRB 050820A, GRB 050922C, GRB 060418, GRB 060526, GRB 060607A, GRB 061007, GRB 061126, GRB 070125, GRB 071010A, GRB 071031, GRB 080310, GRB 080330, GRB 080710, GRB 080810, GRB 081008, GRB 090102, GRB 090926A: all from Kann et al. (2010); GRB 051111: Guidorzi et al. (2006); GRB 060218: Guenther et al. (2006); GRB 060614, GRB 060729, GRB 080413B, GRB 080603A: all from Kann et al. (2011); GRB 060906: Zafar et al. (2011); GRB 060908: de Ugarte Postigo et al. (2011); GRB 071025: Perley et al. (2010); GRB 080319B: Woźniak et al. (2009); GRB 081007: Covino et al. (2013); GRB 081029: Holland et al. (2012); GRB 090426: Nicuesa Guelbenzu et al. (2011); GRB 090510: Nicuesa Guelbenzu et al. (2012); GRB 090618: Cano et al. (2011b); GRB 091018: Wiersema et al. (2012); GRB 091029: Filgas et al. (2012); GRB 091127, GRB 130702A: all from Kann et al. (2016); GRB 100316D: Bufano et al. (2012); GRB 100621A: Greiner et al. (2013); GRB 100814A: Nardini et al. (2014); GRB 101219B: Sparre et al. (2011); GRB 110205A, GRB 110213A: all from Cucchiara et al. (2011); GRB 110918A: Elliott et al. (2013); GRB 111209A: Stratta et al. (2013); GRB 120119A: Morgan et al. (2014); GRB 120404A: Guidorzi et al. (2014); GRB 120422A: Schulze et al. (2014); GRB 120711A: Martin-Carrillo et al. (2014); GRB 120729A, GRB 130215A: all from Cano et al. (2014); GRB 120815A: Krühler et al. (2013); GRB 121024A: Varela et al. (2016); GRB 121217A: Elliott et al. (2014); GRB 130427A: Perley et al. (2014); GRB 130831A: De Pasquale et al. (2016); GRB 130925A: Greiner et al. (2014). References for data sources: (1) Sokolov et al. 1998; (2) Bloom et al. 1998; (3) Castro-Tirado et al. 1998; (4) Galama et al. 1998; (5) Zharikov et al. 1998; (6) Sokolov et al. 1999; (7) Galama et al. 1998; (8) Galama et al. 1999; (9) McKenzie & Schaefer 1999; (10) Clocchiatti et al. 2011; (11) Harrison et al. 1999; (12) Israel et al. 1999; (13) Beuermann et al. 1999; (14) Curran et al. 2008; (15) Sahu et al. 2000; (16) Björnsson et al. 2001; (17) Christensen et al. 2004; (18) Bhargavi & Cowsik 2000; (19) Sagar et al. 2000; (20) Masetti et al. 2000; (21) Rhoads & Fruchter 2001; (22) Jensen et al. 2001; (23) Gaudi 2001; (24) Price et al. 2001; (25) Fynbo et al. 2001; (26) Fynbo et al. 2002; (27) Holland et al. 2003; (28) Mirabal et al. 2003; (29) Pandey et al. 2003; (30) Uemura et al. 2003; (31) de Ugarte Postigo et al. 2005; (32) Fatkhullin et al. 2003; (33) Klose et al. 2004; (34) Pandey et al. 2004; (35) Maiorano et al. 2006; (36) Torii et al. 2003; (37) Bloom et al. 2004; (38) Resmi et al. 2005; (39) Thöne et al. 2007; (40) Klotz et al. 2005; (41) Blustin et al. 2006; (42) Holland et al. 2006; (43) Kann et al. 2010; (44) Resmi et al. 2012; (45) Rykoff et al. 2006; (46) de Pasquale et al. 2007; (47) Cenko et al. 2006; (48) Li et al. 2005; (49) Butler et al. 2006; (50) Guidorzi et al. 2007; (51) Yost et al. 2007; (52) Guidorzi et al. 2006; (53) Ferrero et al. 2006; (54) Sollerman et al. 2006; (55) Mirabal et al. 2006; (56) Brown et al. 2009; (57) Cobb 2006; (58) Molinari et al. 2007; (59) Dai et al. 2007; (60) Thöne et al. 2010; (61) Ziaeepour et al. 2008; (62) Bernardini et al. 2009; (63) Nysewander et al. 2009; (64) Cobb et al. 2006; (65) Della Valle et al. 2006a; (66) Mangano et al. 2007; (67) Xu et al. 2009; (68) Yang et al. 2015; (69) Grupe et al. 2007; (70) Rykoff et al. 2009; (71) Cenko et al. 2009; (72) Covino et al. 2010; (73) Mundell et al. 2007; (74) Gomboc et al. 2008; (75) Perley et al. 2008; (76) Chandra et al. 2008; (77) Updike et al. 2008; (78) Covino et al. 2008; (79) Perley et al. 2010; (80) Krühler et al. 2009b; (81) Modjaz et al. 2009; (82) Littlejohns et al. 2012; (83) Vreeswijk et al. 2013; (84) Bloom et al. 2009; (85) Pandey et al. 2009; (86) Woźniak et al. 2009; (87) Guidorzi et al. 2009; (88) Filgas et al. 2011b; (89) Guidorzi et al. 2011; (90) Perley et al. 2011; (91) Krühler et al. 2009a; (92) Page et al. 2009; (93) Olivares et al. 2015; (94) Yuan et al. 2010; (95) Nardini et al. 2011; (96) Gendre et al. 2010; (97) Nicuesa Guelbenzu et al. 2011; (98) Xin et al. 2011; (99) Thöne et al. 2011; (100) Nicuesa Guelbenzu et al. 2012; (101) Cano et al. 2011b; (102) Page et al. 2011; (103) Rau et al. 2010; (104) Cenko et al. 2011; (105) Wiersema et al. 2012; (106) Virgili et al. 2013; (107) Filgas et al. 2012; (108) Cobb et al. 2010; (109) Olivares et al. 2012; (110) Greiner et al. 2013; (111) Nardini et al. 2014; (112) Sparre et al. 2011; (113) Cucchiara et al. 2011; (114) Ukwatta et al. 2011; (115) Kuroda et al. 2011; (116) Zhao et al. 2011; (117) Volnova et al. 2011; (118) Elliott et al. 2013; (119) Stratta et al. 2013; (120) Kann et al. 2016; (121) Levan et al. 2014; (122) Morgan et al. 2014; (123) Guidorzi et al. 2014; (124) Melandri et al. 2012; (125) Schulze et al. 2014; (126) Martin-Carrillo et al. 2014; (127) Cano et al. 2014; (128) Krühler et al. 2013; (129) Varela et al. 2016; (130) Elliott et al. 2014; (131) Maselli et al. 2014; (132) Becerra et al. 2017; (133) D'Elia et al. 2015; (134) Toy et al. 2016; (135) Volnova et al. 2017; (136) De Pasquale et al. 2016; (137) Gorbovskoy et al. 2015; (138) Greiner et al. 2014.

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Specifically, ${A}_{{\rm{V}}}^{\mathrm{host}}$ is chosen by the following three steps. First, we selected the values obtained by Kann et al. (2006) and Kann et al. (2010) corresponding to the Small Magellanic Cloud (SMC) dust model. Second, when they are not available in Kann et al. (2006) and Kann et al. (2010), the extinction parameters are retrieved from other papers (given as a reference in Table 1) from the SMC dust model. This model is our first choice since it is shown to best describe dust in GRB environments (Kann et al. 2010). When the information is not available for this model, the value of ${A}_{{\rm{V}}}^{\mathrm{host}}$ is obtained from either the Milky Way (MW) or the large Magellanic Cloud (LMC) models. Finally, for a few cases, when the value of ${A}_{{\rm{V}}}^{\mathrm{host}}$ is unavailable for any model, we use the value obtained by fitting a Gaussian to the extinction distribution (see Figure 1), log₁₀(${A}_{{\rm{V}}}^{\mathrm{host}}$) = −0.82 ± 0.41; here −0.82 and 0.41 are the mean value and standard deviation of the Gaussian fit,¹⁶ respectively. After obtaining the value of ${A}_{{\rm{V}}}^{\mathrm{host}}$, we derived the extinction values in any optical band, ${A}_{\lambda }^{\mathrm{host}}$, in the GRB host galaxies following Pei (1992). Figure 2 illustrates the spectral behavior of ${A}_{\lambda }^{\mathrm{host}}/{A}_{{\rm{V}}}^{\mathrm{host}}$ for the three dust models and displays the position of the filters considered in this paper.

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In this section, we describe the fits to the GRB light curves in our 70 bursts. The purpose of this procedure is twofold. First, it allows us to identify different emission phases. Second, the results are used to interpolate the magnitude, in order to compute the CIs, when there are no simultaneous multiwavelength observations available.

The observed magnitude m is obtained from the flux F as

$$\begin{eqnarray}&&m=-2.5{\mathrm{log}}_{10}\left(\displaystyle \frac{{\rm{F}}}{{{\rm{F}}}_{0}}\right),\end{eqnarray} \tag{ 4 }$$

where F₀ is the flux of an object with magnitude zero. Note that in this paper we adopt the AB magnitude system.

After taking into account all the correction factors (our Galaxy and the host galaxy extinction corrections, and the spectral k-correction), we fit the light curves in each band separately with a model of multiple components (Li et al. 2012). The basic model is either a single power law (SPL) or a smoothly broken power law (BKPL) for the flux, resulting in¹⁷

$$\begin{eqnarray}{m}_{{\rm{T}}}(t) & = & -2.5{\mathrm{log}}_{10}({10}^{-0.4{{\rm{m}}}_{{\rm{c}}}}\,{{\rm{t}}}^{-\alpha }+{10}^{-0.4{{\rm{m}}}_{{\rm{G}}}}+{10}^{-0.4{{\rm{m}}}_{{\rm{K}}}}\\ & & +{10}^{-0.4{{\rm{m}}}_{\mathrm{host}}}+{10}^{-0.4{{\rm{m}}}_{\mathrm{GF}}}),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}{m}_{{\rm{T}}}(t) & = & -2.5{\mathrm{log}}_{10}\{{10}^{-0.4{m}_{{\rm{c}}}}{[{(t/{t}_{{\rm{b}},1})}^{{\alpha }_{1}{\omega }_{1}}+{(t/{t}_{{\rm{b}},1})}^{{\alpha }_{2}{\omega }_{1}}]}^{-1/{\omega }_{1}}\\ & & +{10}^{-0.4{m}_{{\rm{G}}}}+{10}^{-0.4{m}_{{\rm{K}}}}+{10}^{-0.4{m}_{\mathrm{host}}}+{10}^{-0.4{m}_{\mathrm{GF}}}\}.\end{eqnarray} \tag{ 6 }$$

Here α, α₁, and α₂ are the temporal slopes, ${t}_{{{\rm{b}}}_{1}}$ is the break time, and ω₁ describes the sharpness of the break (the smaller the value, the smoother the break). For most GRBs, we fix this last parameter to ω₁ = 3. For a few cases (GRB 030226, GRB 050922C, GRB 061007, GRB 080603A) that have a smoother break, we fix it to ω₁ = 1, which improves the fit. Equation (5) can be used for both the light curves with a break and those that exhibit an increasing behavior (e.g., afterglow onset indicated by the dashed line in Figure 3(b)). In addition, m_c is the magnitude constant value, m_G is the extinction magnitude for our Galaxy, m_K is the spectral k-correction, m_host is the extinction magnitude for the host galaxy, and m_GF is the possible contribution from the host galaxy at late time (if identified). Note that m_G, m_K, and m_host are known before fitting.

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We notice that for two bursts (GRB 061126 and GRB 080319B) the optical-afterglow light curves have a steeper decay slope α₁ in the pre-break segment (which might originate from the reverse-shock emission) than the decay slope α₂ in the post-break segment (possibly from the normal decay dominated by the external forward shock). However, due to the limitation of the BKPL model, it is impossible to fit the light curves whose pre-break slope is steeper than the post-break slope. In this case, we adopted the BKPL model function with a negative sharpness of the break ω = −3 (Figure 3(c)).

A double-BKPL light curve (Figure 3(d)) is also expected in some afterglow models. For example, it is theoretically expected that the afterglow light curves may have a shallow segment owing to energy injection at an early time, which then changes into a normal-decay segment when the energy injection is over, and finally steepens owing to a jet break (Zhang et al. 2006). We therefore consider a smooth triple-power-law function (TPL) to fit the light curves (e.g., Liang et al. 2008; Li et al. 2012):

$$\begin{eqnarray}{m}_{{\rm{T}}}(t) & = & -2.5{\mathrm{log}}_{10}\{(({10}^{-0.4{{\rm{m}}}_{{\rm{c}}}}[{({\rm{t}}/{{\rm{t}}}_{{\rm{b}},1})}^{{\alpha }_{1}{\omega }_{1}}\\ & & +{{{(t/{t}_{{\rm{b}},1})}^{{\alpha }_{2}{\omega }_{1}}]}^{-1/{\omega }_{1}})}^{-1/{\omega }_{2}}\\ & & +(({10}^{-0.4{{\rm{m}}}_{{\rm{c}}}}{[{(t/{t}_{{\rm{b}},1})}^{{\alpha }_{1}{\omega }_{1}}+{(t/{t}_{{\rm{b}},1})}^{{\alpha }_{2}{\omega }_{1}}]}^{-1/{\omega }_{1}})({t}_{{\rm{b}},2})\\ & & \times {{{(t/{t}_{{\rm{b}},2})}^{-{\alpha }_{3}}))}^{-1/{\omega }_{2}})}^{-1/{\omega }_{2}}\\ & & +{10}^{-0.4{{\rm{m}}}_{{\rm{G}}}}+{10}^{-0.4{{\rm{m}}}_{{\rm{K}}}}+{10}^{-0.4{{\rm{m}}}_{\mathrm{host}}}\},\end{eqnarray} \tag{ 7 }$$

where ω₂ is the sharpness parameter at the second break time t_b,2.

The fits are performed with the IDL routine "mpfitfun.pro"¹⁸ (Markwardt 2009), which uses the Levenberg–Marquardt algorithm to achieve minimization. To select the best model, we compare the reduced χ² values and choose the one that has a more reasonable value (close to 1). For example, the ${\chi }_{{\rm{r}}}^{2}$ of GRB 990510 (R band) is 31/39 (a little less than 1) for the BKPL model and 169/36 (much larger than 1) for the SPL model. Therefore, we adopt the BKPL as the best model for GRB 990510. Second, if both models have a ${\chi }_{{\rm{r}}}^{2}$ close to 1, our principle is to choose the simpler one (fewer parameters). For instance, for GRB 060908, ${\chi }_{{\rm{r}}}^{2}$ is equal to 38/49 (∼1) for the BKPL model and 52/46 (∼1) for the SPL model. In this case, we choose the SPL model.

For multicomponent light curves (see Section 2.3), we introduce the minimum number of components by eye inspection of the temporal features. If the ${\chi }_{{\rm{r}}}^{2}$ is still much larger than 1, we continue to add more components and redo the fit, until the ${\chi }_{{\rm{r}}}^{2}$ becomes close to 1 (Li et al. 2012). For instance, the value of the ${\chi }_{{\rm{r}}}^{2}$ of GRB 071025 (J band) for the BKPL is 474/39 (much larger than 1), while it is 58/34 (close to 1) for the double-BKPL model. As a result, the double-BKPL model is selected as the best model for this burst. Examples of light-curve fitting with various models or their composition are shown in Figure 4.

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Optical afterglows have complicated light curves with up to eight emission episodes (Kann et al. 2006, 2010; Liang et al. 2006, 2013; Kann et al. 2011; Nardini et al. 2006; Panaitescu & Vestrand 2008, 2011; Li et al. 2012, 2015). A synthetic optical-afterglow light curve is presented in Li et al. (2012). To study the spectral behaviors in these different emission phases, we investigate the temporal evolution of their mean CIs. Here we explain which components are considered and how they are identified based on their temporal and spectral features.

Ia: the prompt optical flares (Prompt Optical); in the very early time of some bursts when the prompt GRB emission is still going on, a highly variable optical emission component may be observed as in GRB 080319B.¹⁹ They are selected such that the observation time is smaller than T90 and the temporal slope²⁰ α > 2.0.

Ib: the early optical flares (Reverse Shock); in a few cases, the early light curves have steep slopes, which likely indicate an early reverse-shock emission component, such as the early phase of GRB 061126. They were selected such that their typical temporal index α ∼ 1.7 and the peak time t_p is a few hundred seconds.

II: an early shallow-decay component (Energy Injection); the afterglow light curve might show an initial shallow-decay segment followed by a normal-decay/post-jet-break segment, which is likely due to energy injection from a long-lasting spinning-down central engine or piling up of flare materials into the blast wave (Liang et al. 2007; Li et al. 2012, 2015). This component is described as the energy injection phase. They are identified by their temporal index α^Shallow < α^Normal with a typical value α^Shallow ∼ 0.5 and with a typical value of break time t_b ∼ 10⁴ s. Here α^Shallow is the temporal index during the shallow-decay segment and α^Normal is the temporal index during the normal-decay phase.

III: the standard afterglow component (Onset/Normal Decay); the light curves sometimes have an early onset rising segment followed by a normal decay. In most of the cases, a lack of observations in the early time lead to only a single normal decay. Afterglow onsets are identified with an early smooth bump with a typical value of peak time t_p of several hundreds of seconds, coupled to the following normal decay with α ∼ 1.2.

IV: the jet-break component (Jet Break); the light curves break into a steeper decay. They are identified with α ∼ −p ∼ 2.5 (Zhang et al. 2006) and a break time ∼10⁵ s. Here p is the electron index.

V: the late optical flares (Flare); the light curves have prompt-like flares during the afterglow phase when the prompt emission is turned off. They indicate the late-time activities of the central engine. The late optical flares are characterized by a very sharp temporal index α < 2.0.

VI: the late re-brightening bumps (Re Bump); the late bumps would emerge at late times, which is distinguished from the onset bump of the early afterglow. Both are likely involved in the jet component that produces the re-brightening bump, which seems to be on-axis and independent of the prompt emission jet component (Liang et al. 2013). They are described by a smooth bump around t_p ∼ 10⁵ s.

VII: the late SN bumps (SN Bump); in some cases, the optical transient light curves at late time show an SN bump. They form a late smooth bump at t_p ∼ 10⁶ s. In addition, we checked that the GRB-SN associations were confirmed in the literature (see Section 5.2.3).

All the assigned emission components in our 70 bursts are presented in Tables 2 and 3 and are marked with arrows in Figures 5 and 6.

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View extended figure (5 panels)

Table 2.  Color Indices of the Golden Sample

GRB	Componenta	Epochb	Modelc	$g-r$	r−i	i−z	J−H	H−Ks
		(ks)		$\overline{\mathrm{mag}}$	$\overline{\mathrm{mag}}$	$\overline{\mathrm{mag}}$	$\overline{\mathrm{mag}}$	$\overline{\mathrm{mag}}$
071025(0)d	Global	0.17–14.9	⋯	⋯	⋯	⋯	0.19 ± 0.14	0.51 ± 0.15
071025(1)	Onset	0.17–0.6	BKPL	⋯	⋯	⋯	0.19 ± 0.25	0.64 ± 0.24
071025(2)	Normal Decay	0.55–1.0	BKPL	⋯	⋯	⋯	0.22 ± 0.10	0.49 ± 0.11
071025(3)	Re Bump	0.99–14.9	BKPL	⋯	⋯	⋯	0.18 ± 0.11	0.45 ± 0.13
071031(0)	Global	0.29–25.7	⋯	0.56 ± 0.03	0.13 ± 0.03	0.09 ± 0.04	0.20 ± 0.06	0.28 ± 0.07
071031(1)	Onset	0.29–1.2	BKPL	0.63 ± 0.02	0.17 ± 0.02	0.09 ± 0.03	0.21 ± 0.05	0.33 ± 0.06
071031(2)	Normal Decay	1.23–4.0	BKPL	0.58 ± 0.04	0.14 ± 0.03	0.08 ± 0.04	0.21 ± 0.05	0.27 ± 0.07
071031(3)	Re Bump	4.05–25.7	BKPL	0.55 ± 0.03	0.12 ± 0.03	0.09 ± 0.04	0.19 ± 0.07	0.22 ± 0.08
080413B(0)	Global	0.34–780.5	⋯	0.14 ± 0.06	0.02 ± 0.06	0.14 ± 0.07	⋯	⋯
080413B(1)	Normal Decay	0.34–90.3	BKPL	0.12 ± 0.06	0.00 ± 0.06	0.15 ± 0.06	⋯	⋯
080413B(2)	Jet Break	90.3–780.5	BKPL	0.30 ± 0.12	0.19 ± 0.10	0.06 ± 0.12	⋯	⋯
080710(0)	Global	0.42–353.1	⋯	0.33 ± 0.01	0.18 ± 0.01	0.19 ± 0.02	0.28 ± 0.02	0.28 ± 0.02
080710(1)	Onset	0.41–2.8	TPL	0.34 ± 0.01	0.19 ± 0.01	0.20 ± 0.01	⋯	⋯
080710(2)	Normal Decay	2.83–7.6	TPL	0.32 ± 0.01	0.17 ± 0.01	0.19 ± 0.01	0.26 ± 0.02	0.31 ± 0.03
080710(3)	Jet Break	7.63–353.1	TPL	0.32 ± 0.01	0.17 ± 0.02	0.18 ± 0.02	0.28 ± 0.02	0.28 ± 0.02
081007(0)	Global	1.12–2500.0	⋯	−0.09 ± 0.22	0.05 ± 0.24	0.02 ± 0.22	0.46 ± 0.07	0.17 ± 0.07
081007(1)	Jet Break	1.12–500.0	BKPL	−0.09 ± 0.22	0.03 ± 0.17	0.02 ± 0.15	0.46 ± 0.07	0.17 ± 0.07
081007(2)	SN Bump	500.0–2500.0	BKPL	⋯	0.08 ± 0.39	0.02 ± 0.44	⋯	⋯
081008(0)	Global	13.65–187.8	BKPL	0.19 ± 0.03	0.32 ± 0.04	0.20 ± 0.04	0.15 ± 0.04	0.00 ± 0.05
081008(1)	Normal Decay	13.65–187.8	BKPL	0.19 ± 0.03	0.32 ± 0.04	0.20 ± 0.04	0.15 ± 0.04	0.00 ± 0.05
081029(0)	Global	0.52–438.7	⋯	1.03 ± 0.03	0.30 ± 0.04	0.19 ± 0.04	0.26 ± 0.09	0.16 ± 0.09
081029(1)	Energy Break	0.52–0.9	BKPL	1.02 ± 0.02	0.28 ± 0.03	0.18 ± 0.03	0.19 ± 0.09	0.11 ± 0.10
081029(2)	Normal Decay	0.94–3.5	BKPL	1.00 ± 0.02	0.27 ± 0.03	0.18 ± 0.03	0.22 ± 0.09	0.14 ± 0.10
081029(3)	Re Bump	3.55–16.3	TPL	1.04 ± 0.03	0.32 ± 0.03	0.20 ± 0.03	0.29 ± 0.07	0.16 ± 0.09
081029(4)	Jet Break	16.26–438.7	TPL	1.04 ± 0.06	0.28 ± 0.07	0.17 ± 0.05	0.22 ± 0.11	0.18 ± 0.07
090426(0)	Global	44.73–139.8	⋯	0.35 ± 0.06	0.05 ± 0.08	0.09 ± 0.12	0.08 ± 0.18	⋯
090426(1)	Energy Break	0.1–0.5	BKPL	⋯	⋯	⋯	⋯	⋯
090426(2)	Normal Decay	0.5–10.0	BKPL	⋯	⋯	⋯	⋯	⋯
090426(3)	Jet Break	10.0–139.8	BKPL	0.36 ± 0.06	0.04 ± 0.09	0.09 ± 0.12	0.08 ± 0.18	⋯
090510(0)	Global	23.13–34.4	⋯	0.33 ± 0.49	0.19 ± 0.41	0.05 ± 0.46	⋯	⋯
090510(1)	Normal Decay	23.13–34.4	PL	0.33 ± 0.49	0.19 ± 0.41	0.05 ± 0.46	⋯	⋯
090926A(0)	Global	73.16–2070.8	⋯	0.19 ± 0.04	0.13 ± 0.04	0.13 ± 0.05	⋯	⋯
090926A(1)	Normal Decay	73.16–200.0	BKPL	0.18 ± 0.04	0.12 ± 0.04	0.13 ± 0.04	⋯	⋯
090926A(2)	Re Bump	200.0–2070.8	BKPL	0.23 ± 0.08	0.18 ± 0.05	0.17 ± 0.07	⋯	⋯
091018(0)	Global	10.97–272.8	⋯	0.12 ± 0.03	0.11 ± 0.04	0.10 ± 0.05	0.09 ± 0.05	0.12 ± 0.08
091018(1)	Energy Break	10.97–19.9	BKPL	0.12 ± 0.03	0.10 ± 0.03	0.10 ± 0.04	0.10 ± 0.05	0.12 ± 0.07
091018(2)	Normal Decay	19.95–272.8	BKPL	0.11 ± 0.05	0.13 ± 0.06	0.08 ± 0.08	0.07 ± 0.08	0.11 ± 0.10
091029(0)	Global	0.31–344.9	⋯	0.25 ± 0.03	0.02 ± 0.03	0.14 ± 0.06	⋯	⋯
091029(0)	Onset	0.31–0.5	BKPL	0.30 ± 0.04	0.09 ± 0.03	0.19 ± 0.05	⋯	⋯
091029(1)	Normal Decay	0.52–4.7	BKPL	0.26 ± 0.03	0.03 ± 0.03	0.15 ± 0.05	⋯	⋯
091029(2)	Re Bump	4.66–344.9	BKPL	0.23 ± 0.03	0.01 ± 0.04	0.13 ± 0.06	⋯	⋯
091127(0)	Global	3.3–4673.7	⋯	0.03 ± 0.02	0.04 ± 0.02	0.02 ± 0.02	⋯	⋯
091127(1)	Energy Break	3.3–27.5	BKPL	0.01 ± 0.01	0.02 ± 0.01	0.01 ± 0.01	⋯	⋯
091127(2)	Normal Decay	27.47–959.0	BKPL	0.19 ± 0.05	0.10 ± 0.05	0.00 ± 0.07	⋯	⋯
091127(3)	SN Bump	959.04–4673.7	BKPL	0.67 ± 0.11	0.43 ± 0.10	0.29 ± 0.16	⋯	⋯
100316D(0)	Global	42.55–7000.0	⋯	0.31 ± 0.06	−0.05 ± 0.06	0.40 ± 0.06	−0.35 ± 0.28	⋯
100316D(1)	Late Bump	42.55–300.0	BKPL	−0.16 ± 0.05	−0.01 ± 0.06	0.03 ± 0.07	−0.27 ± 0.40	⋯
100316D(2)	SN Bump	300.0–7000.0	BKPL	0.51 ± 0.06	−0.07 ± 0.06	0.54 ± 0.06	−0.36 ± 0.27	⋯
100621A(0)	Global	0.26–100.9	⋯	0.98 ± 0.24	0.32 ± 0.17	−0.20 ± 0.15	0.52 ± 0.15	0.67 ± 0.14
100621A(0)	Onset	0.22–0.6	BKPL	⋯	0.15 ± 0.17	0.07 ± 0.13	0.65 ± 0.14	0.49 ± 0.13
100621A(1)	Normal Decay	0.62–2.1	BKPL	0.24 ± 0.25	0.39 ± 0.24	0.11 ± 0.13	0.51 ± 0.16	0.48 ± 0.16
100621A(2)	Re Bump	2.13–100.9	BKPL	1.35 ± 0.24	0.33 ± 0.17	−0.28 ± 0.16	0.48 ± 0.15	0.83 ± 0.13
100814A(0)	Global	0.67–2264.3	⋯	0.08 ± 0.03	0.12 ± 0.04	0.29 ± 0.06	0.15 ± 0.10	0.02 ± 0.16
100814A(1)	Normal Decay	0.67–23.2	BKPL	0.00 ± 0.02	0.07 ± 0.03	0.25 ± 0.05	0.13 ± 0.10	−0.04 ± 0.14
100814A(2)	Re Bump	23.21–2264.3	BKPL	0.19 ± 0.04	0.20 ± 0.05	0.34 ± 0.07	0.22 ± 0.12	0.13 ± 0.20
101219B(0)	Global	29.55–3070.0	⋯	0.29 ± 0.11	0.13 ± 0.12	−0.27 ± 0.19	0.26 ± 0.23	0.34 ± 0.28
101219B(1)	Normal Decay	29.55–500.0	BKPL	0.05 ± 0.09	0.01 ± 0.07	−0.10 ± 0.10	0.24 ± 0.16	0.34 ± 0.28
101219B(2)	SN Bump	500.0–3070.0	BKPL	0.61 ± 0.14	0.25 ± 0.16	−0.41 ± 0.26	0.35 ± 0.59	⋯
110918A(0)	Global	126.39–553.4	⋯	0.18 ± 0.04	0.04 ± 0.04	0.09 ± 0.07	0.33 ± 0.15	0.00 ± 0.20
110918A(1)	Normal Decay	126.39–553.4	BKPL	0.18 ± 0.04	0.04 ± 0.04	0.09 ± 0.07	0.33 ± 0.15	0.00 ± 0.20
111209A(0)	Global	64.49–6241.9	BKPL	−0.03 ± 0.08	−0.04 ± 0.05	−0.10 ± 0.06	0.24 ± 0.21	0.30 ± 0.28
111209A(1)	Late Bump	64.49–1000.0	BKPL	−0.05 ± 0.06	−0.05 ± 0.04	−0.09 ± 0.06	0.22 ± 0.20	0.28 ± 0.26
111209A(2)	SN Bump	1000.0–6241.9	BKPL	0.12 ± 0.20	0.13 ± 0.10	−0.22 ± 0.13	0.48 ± 0.40	0.59 ± 0.68
120711A(0)	Global	21.16–370.1	⋯	0.33 ± 0.15	0.33 ± 0.10	0.08 ± 0.11	0.40 ± 0.19	0.32 ± 0.23
120711A(1)	Normal Decay	21.16–370.1	BKPL	0.33 ± 0.15	0.33 ± 0.10	0.08 ± 0.11	0.40 ± 0.19	0.32 ± 0.23
120815A(0)	Global	0.17–11.1	⋯	0.71 ± 0.03	0.33 ± 0.03	0.17 ± 0.04	⋯	⋯
120815A(0)	Onset	0.17–0.5	BKPL	0.73 ± 0.03	0.32 ± 0.02	0.19 ± 0.03	⋯	⋯
120815A(1)	Normal Decay	0.48–11.1	BKPL	0.71 ± 0.03	0.33 ± 0.03	0.17 ± 0.04	⋯	⋯
121024A(0)	Global	11.09–107.0	⋯	0.72 ± 0.15	0.23 ± 0.09	0.15 ± 0.11	0.34 ± 0.24	0.28 ± 0.26
121024A(1)	Normal Decay	11.09–30.0	BKPL	0.74 ± 0.13	0.22 ± 0.08	0.15 ± 0.08	0.37 ± 0.15	0.21 ± 0.17
121024A(2)	Jet Break	30.0–107.0	BKPL	0.68 ± 0.20	0.23 ± 0.13	0.16 ± 0.16	0.29 ± 0.43	0.43 ± 0.46
121217A(0)	Global	1.12–347.2	⋯	0.54 ± 0.07	0.11 ± 0.06	0.02 ± 0.08	⋯	⋯
121217A(1)	Onset	1.12–1.6	BKPL	0.53 ± 0.06	0.13 ± 0.06	0.09 ± 0.06	⋯	⋯
121217A(2)	Normal Decay	1.59–347.2	BKPL	0.54 ± 0.07	0.11 ± 0.06	0.01 ± 0.08	⋯	⋯
130427A(0)	Global	0.14–3480.0	⋯	⋯	⋯	0.15 ± 0.02	⋯	⋯
130427A(1)	Normal Decay	0.14–500.0	BKPL	⋯	⋯	0.14 ± 0.01	⋯	⋯
130427A(2)	SN Bump	500.0–3480.0	BKPL	⋯	⋯	0.17 ± 0.14	⋯	⋯
130925A(0)	Global	0.94–284.4	BKPL	⋯	−0.48 ± 2.17	−0.06 ± 0.78	0.62 ± 0.34	0.40 ± 0.26
130925A(1)	Flare	0.94–10.0	BKPL	⋯	0.06 ± 2.29	0.13 ± 0.51	0.62 ± 0.26	0.44 ± 0.18
130925A(2)	Normal Decay	10.0–284.4	BKPL	⋯	−1.01 ± 2.05	−0.80 ± 1.82	0.60 ± 1.98	0.18 ± 0.72

Notes.

^aThe identifications of various optical-afterglow emission components. ^bThe time intervals for each component in the observer frame. ^cThe models of the light-curve fits for each component. ^dThe serial number, 0, is denoted as global, and other numbers are the components with time sequence.

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Table 3.  Color Indices of the Silver Sample

GRB	Component	Epoch	Model				Color Indices
		(ks)		$\overline{{UVW}2-{UVM}2}$	$\overline{{UVM}2-{UVW}1}$	$\overline{{UVW}1-U}$	$\overline{U-B}$	$\overline{B-V}$	$\overline{V-R}$	$\overline{R-I}$	$\overline{I-J}$	$\overline{J-H}$	$\overline{H-K}$
970508(0)	Global	25.57–40594.2	⋯	⋯	⋯	⋯	−0.23 ± 0.38	0.32 ± 0.28	0.02 ± 0.23	0.34 ± 0.23	⋯	⋯	⋯
970508(1)	Late Bump	25.57–400.0	BKPL	⋯	⋯	⋯	−0.33 ± 0.38	0.17 ± 0.28	0.05 ± 0.23	0.21 ± 0.23	⋯	⋯	⋯
970508(2)	Jet Break	400.0–1030.0	BKPL	⋯	⋯	⋯	−0.24 ± 0.38	0.19 ± 0.28	0.04 ± 0.23	0.22 ± 0.23	⋯	⋯	⋯
970508(3)	Re Bump	1030.0–40594.2	BKPL	⋯	⋯	⋯	−0.14 ± 0.38	0.48 ± 0.28	−0.01 ± 0.23	0.48 ± 0.23	⋯	⋯	⋯
980425(0)	Global	94.85–46400.0	⋯	⋯	⋯	⋯	2.35 ± 0.21	0.87 ± 0.23	1.25 ± 0.23	0.75 ± 0.21	⋯	⋯	⋯
980425(1)	Late Bump	94.85–1330.0	BKPL	⋯	⋯	⋯	2.15 ± 0.21	0.54 ± 0.23	0.99 ± 0.23	0.73 ± 0.21	⋯	⋯	⋯
980425(2)	SN Bump	1330.0–46400.0	BKPL	⋯	⋯	⋯	2.40 ± 0.21	0.96 ± 0.23	1.32 ± 0.23	0.75 ± 0.21	⋯	⋯	⋯
990510(0)	Global	12.44–2000.0	⋯	⋯	⋯	⋯	⋯	0.09 ± 0.09	⋯	⋯	0.64 ± 0.17	⋯	−0.02 ± 0.31
990510(1)	Normal Decay	12.44–92.6	BKPL	⋯	⋯	⋯	⋯	0.14 ± 0.09	⋯	⋯	0.71 ± 0.17	⋯	−0.02 ± 0.31
990510(2)	Jet Break	92.64–2000.0	BKPL	⋯	⋯	⋯	⋯	0.03 ± 0.09	⋯	⋯	0.56 ± 0.17	⋯	−0.01 ± 0.31
990712(0)	Global	15.25–2990.0	BKPL	⋯	⋯	⋯	⋯	⋯	0.09 ± 0.11	0.63 ± 0.18	⋯	⋯	⋯
990712(1)	Normal Decay	15.25–410.3	BKPL	⋯	⋯	⋯	⋯	⋯	0.07 ± 0.11	0.59 ± 0.18	⋯	⋯	⋯
990712(2)	SN Bump	410.3–2990.0	BKPL	⋯	⋯	⋯	⋯	⋯	0.21 ± 0.11	0.81 ± 0.18	⋯	⋯	⋯
000301C(0)	Global	116.76–992.5	⋯	⋯	⋯	⋯	0.28 ± 0.19	0.11 ± 0.17	0.23 ± 0.15	0.27 ± 0.18	0.41 ± 0.15	⋯	⋯
000301C(1)	Energy Break	116.76–170.1	BKPL	⋯	⋯	⋯	0.33 ± 0.19	0.11 ± 0.17	0.21 ± 0.15	0.29 ± 0.18	0.40 ± 0.15	⋯	⋯
000301C(2)	Flare	170.12–239.7	BKPL	⋯	⋯	⋯	0.26 ± 0.19	0.11 ± 0.17	0.23 ± 0.15	0.21 ± 0.18	0.43 ± 0.15	⋯	⋯
000301C(3)	Normal Decay	239.7–992.5	BKPL	⋯	⋯	⋯	0.27 ± 0.19	0.11 ± 0.17	0.24 ± 0.15	0.31 ± 0.18	0.40 ± 0.15	⋯	⋯
000926(0)	Global	74.48–505.2	⋯	⋯	⋯	⋯	0.70 ± 0.17	0.36 ± 0.15	⋯	⋯	2.90 ± 0.39	⋯	1.10 ± 0.17
000926(1)	Normal Decay	74.48–179.3	BKPL	⋯	⋯	⋯	0.64 ± 0.17	0.38 ± 0.15	⋯	⋯	3.06 ± 0.39	⋯	1.10 ± 0.17
000926(2)	Jet Break	179.28–505.2	BKPL	⋯	⋯	⋯	0.78 ± 0.17	0.34 ± 0.15	⋯	⋯	2.68 ± 0.39	⋯	1.10 ± 0.17
021004(0)	Global	2.09–4550.0	⋯	⋯	⋯	⋯	⋯	0.20 ± 0.08	⋯	⋯	0.17 ± 0.12	⋯	0.04 ± 0.19
021004(1)	Energy Break	2.09–10.6	BKPL	⋯	⋯	⋯	⋯	0.15 ± 0.08	⋯	⋯	0.14 ± 0.12	⋯	0.25 ± 0.19
021004(2)	Flare	10.64–350.0	BKPL	⋯	⋯	⋯	⋯	0.20 ± 0.08	⋯	⋯	0.16 ± 0.12	⋯	0.03 ± 0.19
021004(3)	Normal Decay	350.0–4550.0	BKPL	⋯	⋯	⋯	⋯	0.21 ± 0.08	⋯	⋯	0.25 ± 0.12	⋯	0.06 ± 0.19
030226(0)	Global	16.14–353.7	⋯	⋯	⋯	⋯	0.74 ± 0.12	0.02 ± 0.12	⋯	⋯	0.41 ± 0.17	⋯	0.13 ± 0.11
030226(1)	Normal Decay	16.14–87.5	BKPL	⋯	⋯	⋯	0.78 ± 0.12	0.02 ± 0.12	⋯	⋯	0.44 ± 0.17	⋯	0.16 ± 0.11
030226(2)	Jet Break	87.53–353.7	BKPL	⋯	⋯	⋯	0.68 ± 0.12	0.03 ± 0.12	⋯	⋯	0.38 ± 0.17	⋯	0.10 ± 0.11
030328(0)	Global	4.9–227.5	⋯	⋯	⋯	⋯	0.30 ± 0.13	0.09 ± 0.19	0.03 ± 0.18	0.16 ± 0.25	⋯	⋯	⋯
030328(1)	Energy Break	4.9–21.1	BKPL	⋯	⋯	⋯	0.29 ± 0.13	0.09 ± 0.19	−0.04 ± 0.18	0.34 ± 0.25	⋯	⋯	⋯
030328(2)	Normal Decay	21.05–227.5	BKPL	⋯	⋯	⋯	0.30 ± 0.13	0.10 ± 0.19	0.04 ± 0.18	0.14 ± 0.25	⋯	⋯	⋯
030329(0)	Global	11.17–2860.0	⋯	⋯	⋯	⋯	0.08 ± 0.07	0.10 ± 0.06	⋯	⋯	⋯	0.25 ± 0.19	⋯
030329(1)	Energy Break	4.52–28.6	BKPL	⋯	⋯	⋯	0.23 ± 0.07	−0.04 ± 0.06	⋯	⋯	⋯	0.19 ± 0.19	⋯
030329(2)	Normal Decay	28.55–106.8	BKPL	⋯	⋯	⋯	0.13 ± 0.07	0.00 ± 0.06	⋯	⋯	⋯	0.23 ± 0.19	⋯
030329(3)	Re Bump	106.8–2860.0	BKPL	⋯	⋯	⋯	0.02 ± 0.07	0.18 ± 0.06	⋯	⋯	⋯	0.27 ± 0.19	⋯
050525A(0)	Global	0.07–200.0	⋯	0.24 ± 0.31	−0.41 ± 0.31	0.69 ± 0.31	−0.22 ± 0.34	−0.24 ± 0.29	⋯	⋯	⋯	0.19 ± 0.10	⋯
050525A(1)	Normal Decay	0.07–3.0	BKPL	0.15 ± 0.31	−0.37 ± 0.31	0.62 ± 0.31	−0.27 ± 0.34	−0.05 ± 0.29	⋯	⋯	⋯	0.19 ± 0.10	⋯
050525A(2)	Jet Break	3.0–200.0	BKPL	0.33 ± 0.31	−0.45 ± 0.31	0.78 ± 0.31	−0.15 ± 0.34	−0.47 ± 0.29	⋯	⋯	⋯	0.19 ± 0.10	⋯
050801(0)	Global	0.03–107.3	⋯	1.42 ± 0.46	−0.42 ± 0.36	0.69 ± 0.21	−0.14 ± 0.13	−0.02 ± 0.13	−0.32 ± 0.23	−0.27 ± 0.22	0.85 ± 0.14	⋯	⋯
050801(1)	Energy Break	0.03–0.4	BKPL	1.29 ± 0.46	−0.49 ± 0.36	0.82 ± 0.21	−0.21 ± 0.13	−0.09 ± 0.13	−0.25 ± 0.23	−0.34 ± 0.22	0.85 ± 0.14	⋯	⋯
050801(2)	Normal Decay	0.4–107.3	BKPL	1.49 ± 0.46	−0.38 ± 0.36	0.60 ± 0.21	−0.10 ± 0.13	0.02 ± 0.13	−0.36 ± 0.23	−0.23 ± 0.22	0.85 ± 0.14	⋯	⋯
050820A(0)	Global	0.08–663.3	⋯	⋯	⋯	⋯	0.42 ± 0.18	0.58 ± 0.20	⋯	⋯	0.17 ± 0.13	⋯	0.10 ± 0.15
050820A(1)	Onset	0.08–1.0	BKPL	⋯	⋯	⋯	0.59 ± 0.18	0.98 ± 0.20	⋯	⋯	0.08 ± 0.13	⋯	0.15 ± 0.15
050820A(2)	Normal Decay	1.0–8.9	BKPL	⋯	⋯	⋯	0.61 ± 0.18	0.75 ± 0.20	⋯	⋯	0.20 ± 0.13	⋯	0.14 ± 0.15
050820A(3)	Re Bump	8.9–663.3	BKPL	⋯	⋯	⋯	0.39 ± 0.18	0.53 ± 0.20	⋯	⋯	0.17 ± 0.13	⋯	0.10 ± 0.15
050922C(0)	Global	0.14–606.0	⋯	⋯	⋯	⋯	0.08 ± 0.23	0.15 ± 0.25	0.14 ± 0.25	0.21 ± 0.17	⋯	⋯	⋯
050922C(1)	Energy Break	0.14–8.5	BKPL	⋯	⋯	⋯	0.14 ± 0.23	0.18 ± 0.25	0.12 ± 0.25	0.19 ± 0.17	⋯	⋯	⋯
050922C(2)	Normal Decay	8.51–606.0	BKPL	⋯	⋯	⋯	−0.14 ± 0.23	0.04 ± 0.25	0.21 ± 0.25	0.27 ± 0.17	⋯	⋯	⋯
051111(0)	Global	0.03–89.7	⋯	⋯	⋯	⋯	⋯	0.30 ± 0.13	0.21 ± 0.23	0.20 ± 0.23	⋯	⋯	⋯
051111(1)	Normal Decay	0.03–2.69	BKPL	⋯	⋯	⋯	⋯	0.30 ± 0.13	0.22 ± 0.23	0.18 ± 0.23	⋯	⋯	⋯
051111(2)	Jet Break	2.69–89.7	BKPL	⋯	⋯	⋯	⋯	0.30 ± 0.13	0.18 ± 0.23	0.25 ± 0.23	⋯	⋯	⋯
060218(0)	Global	0.26–2870.0	⋯	−0.49 ± 0.11	0.00 ± 0.13	1.21 ± 0.13	0.49 ± 0.12	−0.21 ± 0.16	⋯	⋯	−0.78 ± 0.10	⋯	0.06 ± 0.30
060218(1)	Late Bump	0.26–171.9	BKPL	−0.26 ± 0.11	−1.10 ± 0.13	0.17 ± 0.13	−0.28 ± 0.12	−0.73 ± 0.16	⋯	⋯	−0.84 ± 0.10	⋯	−0.31 ± 0.30
060218(2)	SN Bump	171.94–7870.0	BKPL	−0.51 ± 0.11	0.02 ± 0.13	1.22 ± 0.13	0.51 ± 0.12	−0.14 ± 0.16	⋯	⋯	−0.78 ± 0.10	⋯	0.06 ± 0.30
060418(0)	Global	0.08–101.4	⋯	⋯	⋯	⋯	⋯	−0.07 ± 0.21	⋯	⋯	−0.05 ± 0.16	⋯	0.34 ± 0.14
060418(1)	Onset	0.08–1.0	BKPL	⋯	⋯	⋯	⋯	−0.07 ± 0.21	⋯	⋯	−0.14 ± 0.16	⋯	0.38 ± 0.14
060418(2)	Normal Decay	0.2–101.4	BKPL	⋯	⋯	⋯	⋯	−0.07 ± 0.21	⋯	⋯	0.00 ± 0.16	⋯	0.32 ± 0.14
060526(0)	Global	0.06–893.5	⋯	⋯	⋯	⋯	⋯	0.66 ± 0.15	⋯	⋯	0.14 ± 0.13	⋯	0.27 ± 0.27
060526(1)	Normal Decay	0.06–96.6	BKPL	⋯	⋯	⋯	⋯	0.64 ± 0.15	⋯	⋯	0.17 ± 0.13	⋯	0.24 ± 0.27
060526(2)	Jet Break	96.63–893.5	BKPL	⋯	⋯	⋯	⋯	0.70 ± 0.15	⋯	⋯	0.09 ± 0.13	⋯	0.34 ± 0.27
060607A(0)	Global	0.07–25.5	⋯	⋯	⋯	⋯	⋯	0.62 ± 0.26	⋯	⋯	⋯	⋯	⋯
060607A(1)	Onset	0.07–0.3	BKPL	⋯	⋯	⋯	⋯	0.67 ± 0.26	⋯	⋯	⋯	⋯	⋯
060607A(2)	Normal Decay	0.3–2.8	BKPL	⋯	⋯	⋯	⋯	0.62 ± 0.26	⋯	⋯	⋯	⋯	⋯
060607A(3)	Flare	1.42–25.5	BKPL	⋯	⋯	⋯	⋯	0.58 ± 0.26	⋯	⋯	⋯	⋯	⋯
060614(0)	Global	1.55–1280.0	⋯	0.39 ± 0.64	−0.59 ± 0.57	0.87 ± 0.33	0.09 ± 0.28	0.03 ± 0.28	−0.17 ± 0.32	−0.04 ± 0.26	0.32 ± 0.17	⋯	⋯
060614(1)	Late Bump	1.55–1280.0	BKPL	0.39 ± 0.64	−0.59 ± 0.57	0.87 ± 0.33	0.09 ± 0.28	0.03 ± 0.28	−0.17 ± 0.32	−0.04 ± 0.26	0.32 ± 0.17	⋯	⋯
060729(0)	Global	0.12–2576.8	BKPL	0.26 ± 0.22	−0.05 ± 0.24	⋯	⋯	⋯	0.50 ± 0.31	⋯	⋯	⋯	⋯
060729(1)	Energy Break	0.12–60.0	BKPL	0.19 ± 0.22	−0.25 ± 0.24	⋯	⋯	⋯	0.40 ± 0.31	⋯	⋯	⋯	⋯
060729(2)	Normal Decay	50.04–2576.8	BKPL	0.29 ± 0.22	0.06 ± 0.24	⋯	⋯	⋯	0.55 ± 0.31	⋯	⋯	⋯	⋯
060908(0)	Global	0.06–34343.9	⋯	⋯	⋯	⋯	0.22 ± 0.27	0.10 ± 0.33	⋯	⋯	0.19 ± 0.31	⋯	0.01 ± 0.09
060908(1)	Normal Decay	0.06–34343.9	PL	⋯	⋯	⋯	0.22 ± 0.27	0.10 ± 0.33	⋯	⋯	0.19 ± 0.31	⋯	0.01 ± 0.09
061007(0)	Global	0.03–149.7	⋯	⋯	⋯	⋯	⋯	0.14 ± 0.26	0.67 ± 0.26	⋯	⋯	⋯	⋯
061007(1)	Onset	0.03–0.1	BKPL	⋯	⋯	⋯	⋯	0.15 ± 0.26	0.69 ± 0.26	⋯	⋯	⋯	⋯
061007(2)	Normal Decay	0.14–149.7	BKPL	⋯	⋯	⋯	⋯	0.14 ± 0.26	0.67 ± 0.26	⋯	⋯	⋯	⋯
061126(0)	Global	0.04–1300.0	⋯	⋯	−0.13 ± 0.87	0.47 ± 0.64	0.18 ± 0.19	−0.04 ± 0.16	⋯	⋯	0.46 ± 0.18	⋯	0.48 ± 0.12
061126(1)	Reverse Shock	0.04–1.0	BKPL	⋯	−0.13 ± 0.87	0.47 ± 0.64	0.33 ± 0.19	0.03 ± 0.16	⋯	⋯	0.32 ± 0.18	⋯	0.30 ± 0.12
061126(2)	Normal Decay	1.06–1300.0	BKPL	⋯	−0.13 ± 0.87	0.47 ± 0.64	0.16 ± 0.19	−0.05 ± 0.16	⋯	⋯	0.47 ± 0.18	⋯	0.50 ± 0.12
070125(0)	Global	2.06–106.7	⋯	0.30 ± 0.29	−0.40 ± 0.29	0.77 ± 0.20	−0.09 ± 0.14	⋯	⋯	0.40 ± 0.10	0.11 ± 0.21	⋯	−0.09 ± 0.35
070125(1)	Normal Decay	2.06–5.2	PL	0.33 ± 0.29	−0.44 ± 0.29	0.76 ± 0.20	−0.09 ± 0.14	⋯	⋯	0.40 ± 0.10	0.11 ± 0.21	⋯	−0.09 ± 0.35
070125(2)	Re Bump	5.2–106.7	BKPL	0.11 ± 0.29	−0.21 ± 0.29	0.86 ± 0.20	−0.06 ± 0.14	⋯	⋯	0.40 ± 0.10	0.11 ± 0.21	⋯	−0.07 ± 0.35
071010A(0)	Global	0.12–3523.2	⋯	⋯	⋯	⋯	0.12 ± 0.11	0.42 ± 0.06	⋯	⋯	0.65 ± 0.19	⋯	0.30 ± 0.14
071010A(1)	Onset	0.12–1.0	BKPL	⋯	⋯	⋯	0.14 ± 0.11	0.65 ± 0.06	⋯	⋯	0.65 ± 0.19	⋯	0.23 ± 0.14
071010A(2)	Normal Decay	1.0–36.9	BKPL	⋯	⋯	⋯	0.18 ± 0.11	0.47 ± 0.06	⋯	⋯	0.65 ± 0.19	⋯	0.43 ± 0.14
071010A(3)	Flare	36.91–3523.2	BKPL	⋯	⋯	⋯	0.05 ± 0.11	0.30 ± 0.06	⋯	⋯	0.65 ± 0.19	⋯	0.16 ± 0.14
080109(0)	Global	70.0–2780.0	⋯	⋯	⋯	⋯	1.12 ± 0.15	0.80 ± 0.09	⋯	⋯	0.37 ± 0.15	⋯	−0.08 ± 0.19
080109(1)	Late Bump	70.0–541.1	BKPL	⋯	⋯	⋯	0.71 ± 0.15	0.63 ± 0.09	⋯	⋯	0.26 ± 0.15	⋯	−0.08 ± 0.19
080109(2)	SN Bump	541.09–2780.0	BKPL	⋯	⋯	⋯	1.34 ± 0.15	0.90 ± 0.09	⋯	⋯	0.43 ± 0.15	⋯	−0.07 ± 0.19
080310(0)	Global	0.08–300.0	⋯	⋯	⋯	⋯	⋯	0.34 ± 0.11	⋯	⋯	0.23 ± 0.10	⋯	0.24 ± 0.15
080310(1)	Onset	0.08–0.8	BKPL	⋯	⋯	⋯	⋯	0.26 ± 0.11	⋯	⋯	0.15 ± 0.10	⋯	0.29 ± 0.15
080310(2)	Normal Decay	0.74–2.0	BKPL	⋯	⋯	⋯	⋯	0.36 ± 0.11	⋯	⋯	0.11 ± 0.10	⋯	0.29 ± 0.15
080310(3)	Re Bump	2.0–300.0	BKPL	⋯	⋯	⋯	⋯	0.35 ± 0.11	⋯	⋯	0.29 ± 0.10	⋯	0.21 ± 0.15
080319B(0)	Global	0.71–209.8	⋯	0.50 ± 0.15	−0.63 ± 0.15	0.78 ± 0.14	−0.01 ± 0.11	−0.05 ± 0.12	0.07 ± 0.15	0.13 ± 0.13	0.07 ± 0.05	0.09 ± 0.09	0.04 ± 0.09
080319B(1)	Revese Shock	0.01–1.7	BKPL	0.59 ± 0.15	−0.92 ± 0.15	1.05 ± 0.14	0.02 ± 0.11	0.09 ± 0.12	0.43 ± 0.15	−0.12 ± 0.13	−0.02 ± 0.05	0.13 ± 0.09	0.15 ± 0.09
080319B(2)	Normal Decay	1.68–209.8	BKPL	0.48 ± 0.15	−0.55 ± 0.15	0.73 ± 0.14	0.01 ± 0.11	−0.09 ± 0.12	−0.05 ± 0.15	0.22 ± 0.13	0.12 ± 0.05	0.06 ± 0.09	0.02 ± 0.09
080330(0)	Global	0.09–116.6	⋯	⋯	⋯	1.65 ± 0.34	−0.28 ± 0.26	−0.04 ± 0.29	0.17 ± 0.28	⋯	⋯	⋯	−0.31 ± 0.14
080330(1)	Onset	0.09–0.46	BKPL	⋯	⋯	1.64 ± 0.34	−0.25 ± 0.26	−0.11 ± 0.29	0.29 ± 0.28	⋯	⋯	⋯	−0.34 ± 0.14
080330(2)	Normal Decay	0.46–116.6	BKPL	⋯	⋯	1.66 ± 0.34	−0.29 ± 0.26	−0.01 ± 0.29	0.12 ± 0.28	⋯	⋯	⋯	−0.30 ± 0.14
080603A(0)	Global	0.11–350.4	⋯	⋯	⋯	⋯	⋯	0.39 ± 0.45	⋯	⋯	1.50 ± 0.42	⋯	0.30 ± 0.23
080603A(1)	Onset	0.11–2.0	BKPL	⋯	⋯	⋯	⋯	0.91 ± 0.45	⋯	⋯	1.53 ± 0.42	⋯	0.17 ± 0.23
080603A(2)	Normal Decay	2.0–350.4	BKPL	⋯	⋯	⋯	⋯	0.25 ± 0.45	⋯	⋯	1.50 ± 0.42	⋯	0.34 ± 0.23
080607(0)	Global	0.04–7.5	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1.82 ± 0.29	⋯	1.32 ± 0.14
080607(1)	Onset	0.04–0.2	BKPL	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1.56 ± 0.29	⋯	1.04 ± 0.14
080607(2)	Normal Decay	0.2–1.0	BKPL	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1.95 ± 0.29	⋯	1.53 ± 0.14
080607(3)	Re Bump	1.0–7.5	BKPL	⋯	⋯	⋯	⋯	⋯	⋯	⋯	1.92 ± 0.29	⋯	1.28 ± 0.14
080810(0)	Global	0.04–497.9	⋯	⋯	⋯	⋯	⋯	0.38 ± 0.29	0.45 ± 0.19	0.19 ± 0.14	⋯	⋯	⋯
080810(1)	Onset	0.04–0.3	BKPL	⋯	⋯	⋯	⋯	0.32 ± 0.29	0.54 ± 0.19	0.21 ± 0.14	⋯	⋯	⋯
080810(2)	Normal Decay	0.3–497.9	BKPL	⋯	⋯	⋯	⋯	0.39 ± 0.29	0.45 ± 0.19	0.19 ± 0.14	⋯	⋯	⋯
090102(0)	Global	0.04–264.6	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.66 ± 0.09
090102(1)	Onset	0.04–0.1	BKPL	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.48 ± 0.09
090102(2)	Normal Decay	0.1–2.3	BKPL	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.75 ± 0.09
090102(3)	Re Bump	2.28–264.6	BKPL	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	0.71 ± 0.09
090618(0)	Global	0.12–155.5	BKPL	0.00 ± 0.40	−0.17 ± 0.45	1.28 ± 0.37	−0.34 ± 0.14	0.21 ± 0.20	−0.20 ± 0.17	⋯	⋯	⋯	⋯
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Once the fits are obtained, one can derive the CIs. They are defined as the magnitude difference between two filters,

$$\begin{eqnarray}&&\mathrm{CI}=m({\lambda }_{2})-m({\lambda }_{1}),\end{eqnarray} \tag{ 8 }$$

where λ₂ < λ₁ is required so that the smaller the CI, the bluer (or hotter) the optical afterglow. In this paper, we uniquely focus on the CI between two adjacent bands.²¹ CI can be affected by optical extinction, which can be characterized by the color excess (CE), defined by the difference between the observed color index (CI_O) and the intrinsic color index (CI_T)

$$\begin{eqnarray}&&\mathrm{CE}={\mathrm{CI}}_{{\rm{O}}}-{\mathrm{CI}}_{{\rm{T}}}=A({\lambda }_{2})-A({\lambda }_{1}).\end{eqnarray} \tag{ 9 }$$

2.4.1. Definition of the Golden and Silver Samples

To compute the CIs, simultaneous observations or interpolated values are required. We also note that different telescopes have different photometric systems (i.e., different light filters and spectral response functions), which are either the common Johnson–Cousins UBVRI photometry system²² or the Sloan Digital Sky Survey (SDSS) ugriz photometry system.²³ Instead of converting one system into another with empirical correlations (e.g., Jordi et al. 2006), we group the bursts by photometric systems, which are studied separately.

Therefore, we sort all GRBs into two samples based on the photometric system used for the observation. This also naturally separates bursts for which interpolation is required from those for which it is not.

1. Golden sample: The 25 bursts have simultaneous observations in multiple filters in the optical/IR bands (g, r, i, z, J, H, K_s) for a wide time window. They are mostly observed by the Gamma-Ray Burst Optical/NIR Detector (GROND; Greiner et al. 2008), with some events contributed from observations of the RAPid Telescopes for Optical Response (RAPTOR; Wren et al. 2010) and the Peters Automated IR Imaging Telescope (PAIRITEL; Bloom et al. 2006). These define our Golden sample. Obtaining the CI²⁴ for these bursts is straightforward. The afterglow multicolor light curves and the time evolution of CIs are displayed in Figure 5.

2. Silver sample: The 45 remaining bursts in our sample do not have simultaneous observations in multiple filters but have instead multiple observations in different filters from different optical instruments at different times. For those bursts, we use the light-curve fits (Section 2.3) and interpolate the missing data points to obtain concurrent values in different filters (UVW2, UVM2, UVW1, U, B, V, R, I, J, H and K). This group of bursts defines the Silver sample. It has wider observed energy bands than the Golden sample. We try to include a maximum number of possible multiple data to represent the whole light curve. The multicolor light curves²⁵ and the time evolution of CIs for each burst are shown in Figure 6.

The most accurate CIs are obtained in cases where there are simultaneous observations in multiple filters. This is the case for the bursts in the Golden sample. For many of these bursts there are additional nonsimultaneous data as well, which are discarded in order to keep the sample pure. Three criteria are applied to consider whether to add the additional data sources into the Golden sample: (i) there is at least an order-of-magnitude extension in the duration of observation; (ii) the number of data points should be greater than (or at least equal to) that in the original data source; (iii) at least three new bands have good observational data. Most cases do not satisfy these criteria. However, in two particular cases, GRB 090426 and GRB 130427A, there is a significant amount of additional data. We therefore use these cases to compare the results from the Golden and Silver samples.

For example, we add the R-band data to the multicolor light curves in the Golden sample for GRB 090426 (Figure 5). It provides a much earlier observation that can be compared to the GROND griz observation.

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View extended figure (8 panels)

2.4.2. Methods to Determine the Temporal Behavior of Color Indices

We further define sets of CIs. The first one contains time-resolved data (Data Set I). The second set is made of component-wise averaged CIs (Data Set II). Finally, the mean CIs over the entire burst duration give Data Set III. Hereinafter all the analysis (such as the figures and the tables) will be marked with which sets of data are used.

According to Šimon et al. (2001), CIs should not be derived from fits to the light curve. They claimed that any fits to the data would distort the CIs. The Golden sample gives CIs directly, while the Silver sample CIs are obtained with a fit to the light curve. Studying the difference between Golden and Silver samples allows us to check the consistency between our results and that of Šimon et al. (2001).

To account for the missing data points, two methods can be employed: (i) interpolation of the light curve from the fits (applied to the Silver sample), and (ii) averaging of existing data points as in Šimon et al. (2001). Interpolating missing data points results in large uncertainties but has the advantage of preserving a maximum number of simultaneous CIs. If the afterglow emission is smooth enough, the uncertainties should not be too large. On the other hand, averaging data results in more accurate estimation of the CIs but strongly reduces their number. As pointed out by Šimon et al. (2001), averaging is also not reliable when light curves have rapid changes. Interestingly, after comparing both methods, we find that the results are consistent. We note that our Set II, which is obtained by averaging the CIs on each afterglow component, is similar to averages in arbitrary time intervals, as in Šimon et al. (2001), except that, in our paper, the time intervals are imposed by expected physical variations of the light curves.

We investigate the correlation of the CIs from GRB 130427A using these two methods. This burst has a well-observed afterglow with a great number of data points. In Figure 7, we compare the CIs in the same observation duration between the Golden and the Silver samples. The CIs from the Golden sample are derived directly from RAPTOR (Vestrand et al. 2014) with simultaneous observations in multiple filters, while the CIs from the Silver sample are obtained by an interpolation method from the RATIR (Becerra et al. 2017), which in general do not have simultaneous observation, compared with the GROND observation in the corresponding period. We find that the data cluster around the equal line, which indicates that the results of both methods are consistent with each other.

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2.4.3. Determination of Time Evolution

Following the definition of time evolution outlined in Section 2.4.3, we find that there are 20 out of 25 GRBs in the Golden sample that have at least more than two data points that are variable CIs. This indicates that the evolution of the CIs in the afterglow is very common. GRB 080413B is taken as an example, and its CIs as functions of time are displayed in Figure 12(a). The variable CIs are indicated by arrows. We note that the evolution of CIs tends to appear at a late time. In Figure 12(b), we compare the distributions of constant and variable CIs. We see that the nonvarying CIs are distributed similarly from one color to the others. Fitting the histograms by a Gaussian function, one has g–r = 0.15 ± 0.14 (mean ± standard deviations), r–i = 0.08 ± 0.09, i–z = 0.13 ± 0.10, J–H = 0.23 ± 0.11, and H–K_s = 0.25 ± 0.17, respectively.

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Instead, the group of varying CIs shows a broad distribution for different colors, possibly indicating different underlying phenomena. Table 5 summarizes the total number of constant/variable CIs. In the 25 GRBs of the Golden sample, 4039 out of 4212 (95.9%) values for all five CIs in all the time bins are constant, while only 173 (4.1%) are variable. This indicates that during most of the observed intervals, the CIs are constant, even for the 20 bursts that have a change (a small number of data points) in them.

We investigate the connection between the values of CIs and the afterglow light-curve components to determine their temporal evolution from one component to another. The components were identified in Section 2.4. We calculate the mean value of CIs for each emission component.²⁸ The results are presented in Table 2 for the Golden sample and in Table 3 for the Silver sample.

In Figure 13(a), we show the distribution of CIs among various optical emission components (Data Set II, regardless of the bursts and CIs) and the distribution of ΔCI (results from Gaussian fits are summarized in Table 6) between one component and the following one (Figures 13(b)). ΔCI is defined by

$$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{CI}=\mathrm{CI}({t}_{2})-\mathrm{CI}({t}_{1}),\end{eqnarray} \tag{ 10 }$$

where t₂ > t₁. This means that an increasing CI, with a red-to-blue color change, corresponds to ΔCI > 0; in contrast, ΔCI < 0 represents a decrease in CI with a blue-to-red color change.

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We defined the global sample as the set of all CIs from both the Golden and Silver samples. We find that various emission components generally have similar distributions, with typical values ∼0.2, except the SN component, which presents a different ΔCI distribution. Comparing the CIs, we find that the distributions are consistent, though the number of data points is small. Note, however, that the distribution is very broad.

We present in Figure 14 the CI–CI correlations between consecutive components, regardless of the CIs. We compute the correlation index r in all cases and find that it is systematically close to 1. The two smallest values are the transition to an SN component with r = 0.57 and a flare component with r = 0.60. In addition, a linear fit to the slope for most correlations is also close to 1. This indicates that the CIs do not have a significant change between consecutive components. The fit results, along with the Spearman correlation coefficient r and the chance probability $\hat{p}$, are reported in Table 6. This table also gives the number of CIs per emission component and the distribution range of ΔCI.

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We find that 263 of 306 (85.9%) pair components satisfy the criterion that the CI does not change. The large fraction of CI without significant time variation for both the Golden and Silver samples suggests that the emission mechanism of the various components in the optical transient may stay the same. This is consistent with the prediction of the external shock afterglow model, for which the emission is dominated by synchrotron radiation, provided that the observation bands do not cover the spectral breaks.

Only 43 of the 306 (14.1%) pair components have a significant difference. For these pairs, we found that 18 include SN components, 5 include revese-shock components, and 5 include flares (see Table 5). The Golden and Silver samples show similar results. In those pair components that include an SN, 36% show variation. Likewise, 20% of the pair components including flares show variation. This makes the SN and flare components the ones that exhibit the most common variations. This is consistent with the observational evidence that the optical transient spectrum typically changes during the associated SN and flares (e.g., Šimon et al. 2001, 2004; Zhang et al. 2006; Mu et al. 2016; Geng et al. 2016 2017, and references therein). We also found a few cases of CI variation for the jet-break components. This is noteworthy since a jet break is believed to be purely dynamical. Therefore, CI evolution is not expected. Here, one should bare in mind the possibility that we could have identified the wrong component. For example, we could have identified a flare as a re-brightening bump if the flare is weak (GRB 021004, GRB 000301C, GRB 060707A). This is because they have similar light curves. Another possibility is that the interpolation procedure for the Silver sample introduces uncertainties affecting the determination of the CI.

There is a natural connection between the CIs and the spectral indices. Assuming that the SED of the afterglow in the considered bands is well described by an intrinsically single power law, the spectral flux density (erg cm⁻² s⁻¹ Hz⁻¹) for the afterglow is described by Equation (2). The relation between the flux and the observed magnitude is given by Equation (4), so we can write the CI, which is expressed from the flux density as

$$\begin{eqnarray}&&\mathrm{CI}=-2.5{\mathrm{log}}_{10}\left(\displaystyle \frac{{F}_{{\nu }_{2}}}{{F}_{{\nu }_{1}}}\right)+{\rm{C}},\end{eqnarray} \tag{ 11 }$$

where C = ZP₂–ZP₁, and ZP_i = 2.5log₁₀(${F}_{{\nu }_{i},0}$) is the zero-point of various bands i. Therefore, the spectral index β_o can be connected to the CI as

$$\begin{eqnarray}&&\mathrm{CI}=2.5{\beta }_{o}{\mathrm{log}}_{10}\left(\displaystyle \frac{{\nu }_{2}}{{\nu }_{1}}\right).\end{eqnarray} \tag{ 12 }$$

According to Equation (12), there exists a linear relation between CIs and spectral indices. Therefore, CIs provide a probe to investigate the spectral properties of the optical afterglow.

Using Equation (12), we derive the index ${\beta }_{o}^{\mathrm{CI}}$ using constant CIs of Data Set I of the Golden sample (see Table 7 in the Appendix) and Data Set II of both the Golden and Silver samples during the normal-decay phase. We show the distribution of ${\beta }_{o}^{\mathrm{CI}}$ in Figure 15. The distribution is well fitted by a Gaussian function, and ${\beta }_{o}^{\mathrm{CI}}$ = 0.68 ± 0.60 for constant CIs of Data Set I and ${\beta }_{o}^{\mathrm{CI}}$ = 0.68 ± 0.68 for the Data Set II during the normal-decay phase, which are consistent with each other.

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The distributions are consistent with the predictions of the external shock model, with typical β_o around 0.75 (Mészáros & Rees 1997; Sari et al. 1998; Zhang et al. 2006). The spectral indices inconsistent with the external shock model are obtained from the high-energy bands, which could point toward a stronger reddening. We note that according to the external shock model, ${\beta }_{o}^{\mathrm{CI}}$ should cluster between 0.5 and 1.0. However, we found that a small fraction of data deviate from this range, which have large error bars, and thus they cannot accurately constrain the calculation of CI. Then this uncertainty propagates to the calculation of spectral index.

According to the standard synchrotron afterglow model, radiation is produced by electrons distributed with a power-law index p. Therefore, the CIs can be derived from the following equations for both a constant-density interstellar medium (ISM) and a wind-like medium (wind):

• (i)  
  Fast cooling:

  $$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{CI}=\left\{\begin{array}{l}-2.5{\mathrm{log}}_{10}{\left(\displaystyle \frac{{\nu }_{2}}{{\nu }_{m}}\right)}^{-\displaystyle \frac{(p-1)}{2}},\\ (\mathrm{from}\ {\nu }_{2}\gt {\nu }_{m}\gt {\nu }_{1}\gt {\nu }_{c}\ \mathrm{to}\ {\nu }_{m}\gt {\nu }_{2}\gt {\nu }_{1}\gt {\nu }_{c})\\ -2.5{\mathrm{log}}_{10}{\left(\displaystyle \frac{{\nu }_{1}}{{\nu }_{m}}\right)}^{-\displaystyle \frac{(p-1)}{2}},\\ (\mathrm{from}\ {\nu }_{2}\gt {\nu }_{1}\gt {\nu }_{m}\gt {\nu }_{c}\ \mathrm{to}\ {\nu }_{2}\gt {\nu }_{m}\gt {\nu }_{1}\gt {\nu }_{c})\\ -2.5{\mathrm{log}}_{10}{\left(\displaystyle \frac{{\nu }_{2}}{{\nu }_{1}}\right)}^{-\displaystyle \frac{(p-1)}{2}},\\ (\mathrm{from}\ {\nu }_{2}\gt {\nu }_{1}\gt {\nu }_{m}\gt {\nu }_{c}\ \mathrm{to}\ {\nu }_{m}\gt {\nu }_{2}\gt {\nu }_{1}\gt {\nu }_{c})\end{array}\right.\end{eqnarray} \tag{ 13 }$$
• (ii)  
  Slow cooling:

  $$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{CI}=\left\{\begin{array}{l}-2.5{\mathrm{log}}_{10}{\left(\displaystyle \frac{{\nu }_{2}}{{\nu }_{c}}\right)}^{-\displaystyle \frac{1}{2}},\\ (\mathrm{from}\ {\nu }_{2}\gt {\nu }_{c}\gt {\nu }_{1}\gt {\nu }_{m}\ \mathrm{to}\ {\nu }_{c}\gt {\nu }_{2}\gt {\nu }_{1}\gt {\nu }_{m})\\ -2.5{\mathrm{log}}_{10}{\left(\displaystyle \frac{{\nu }_{1}}{{\nu }_{c}}\right)}^{-\displaystyle \frac{1}{2}},\\ (\mathrm{from}\ {\nu }_{2}\gt {\nu }_{1}\gt {\nu }_{c}\gt {\nu }_{m}\ \mathrm{to}\ {\nu }_{2}\gt {\nu }_{c}\gt {\nu }_{1}\gt {\nu }_{m})\\ -2.5{\mathrm{log}}_{10}{\left(\displaystyle \frac{{\nu }_{2}}{{\nu }_{1}}\right)}^{-\displaystyle \frac{1}{2}},\\ (\mathrm{from}\ {\nu }_{2}\gt {\nu }_{1}\gt {\nu }_{c}\gt {\nu }_{m}\ \mathrm{to}\ {\nu }_{c}\gt {\nu }_{2}\gt {\nu }_{1}\gt {\nu }_{m})\end{array}\right.\end{eqnarray} \tag{ 14 }$$

  Here ν_c is the cooling frequency, ν_m is the minimum frequency, and ν₂ and ν₁ are the frequencies of adjacent bands. In the fast cooling regime, electrons at the injection Lorentz factor have time to lose their energy by emitting synchrotron radiation, while they do not in the slow cooling regime.

Next, we use the observed CIs (Data Set I of the Golden sample) to derive the electron power-law indices p^CI using Equations (13)–(14). In Figure 16, we present the distribution of p^CI, fit by a Gaussian. The mean values with their standard deviations are p^CI = 2.43 ± 1.06 for slow cooling (ν_c > ν₂ > ν₁ > ν_m) and p^CI = 1.43 ± 1.06 for fast/slow cooling (ν₂ > ν₁ > max(ν_c, ν_m)). We find that the slow cooling scenario is consistent with the theoretical predictions for relativistic shocks, p ∼ 2.5. A wide range of the electron index is obtained, which is consistent with previous studies (Shen et al. 2006; Liang et al. 2008).

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Next, we use the closure relations to compute the electron index p from the temporal index α of the normal decay, which was obtained by fitting the light curve, or the spectral index β_o. We then further estimate the expected value of the color indices CI_th (Data Set III of the Golden sample) using Equations (13)–(14). Figure 17 displays the correlation between the CI and CI_th for these spectral regimes. Their distributions are also shown with the best Gaussian fits CI_obs = 0.16 ± 0.14 for observed CI and CI_th = 0.13 ± 0.07 for theoretical CI.

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We find that the data smoothly cluster around the equal line, and both observed CI and theoretical CI_th have similar distributions, consistent with the afterglow synchrotron radiation model with an intrinsic power-law decay.

We also find that a small part of CIs are far away from the equal line, which implies that these data are inconsistent with the model. These CIs are mainly derived from GRB 081029 and GRB 100621A (Figure 17). It is interesting that both GRB 081029 and GRB 100621A present a light curve with re-brightening bumps (see Figure 5). The late re-brightening bumps could come from different emission components. Overlapping bumps could alter the CIs' estimation. However, we also find that CIs preceding the bump are still inconsistent with the model. To investigate this further, we searched all the bursts in the Golden sample to find more cases presenting clear late re-brightening bumps in the light curves. We find that GRB 071025, GRB 091029, and GRB 100814A are consistent with the theoretical values and present a late re-brightening. The results show that the afterglow emission for both GRB 081029 and GRB 100621A is likely different from other bursts.

In Figure 18, we show color–color diagrams using the Golden sample (Data Set I) to investigate the evolution of CI shift. We calculate the distance of data points to the equal line and the difference value between the two CIs. We find that most of the CIs cluster around the equal line and the typical difference value between two CIs is close to 0. For the distances, one has (g–r)–(r–i) = −0.03 ± 0.07, (r–i)–(i–z) = 0.00 ± 0.06, and (J–H)–(H–K_s) = 0.00 ± 0.14, respectively. For their differences, one has (g–r)–(r–i) = −0.04 ± 0.10, (r–i)–(i–z) = 0.01 ± 0.09, and (J–H)–(H–K_s) = 0.00 ± 0.20, respectively.

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This result indicates that the CI shift is not significant, and the intrinsic reddening inside their host galaxies must be quite similar and relatively small for these events. This is also expected from the narrow distributions in wavelength of the ugriz filters. This result also implies that the afterglow spectral shapes are similar from one to another and can be described by a smooth power-law spectral decay, with no bumps or strong lines for these bands. These results are consistent with the finding of Šimon et al. (2004).

Below we outline three possible explanations for the CI variation, and in Section 5.2.4 we compare the prediction to the data.

5.2.1. The Cooling Frequency Crosses the Studied Energy Bands

5.2.2. Effects of Host Extinction

The host extinction may also cause the CI change (e.g., Waxman & Draine 2000; Morgan et al. 2014). After being eventually destroyed by the initial intense radiation, dust could reform at an early time near the burst region, changing the host extinction (Perna & Lazzati 2002). This leads to two temporal changes in the CI. During the early afterglow phase (Phase I), dust might be destroyed, and the CI can be described by a red-to-blue change, implying an increase in the CI with ΔCI > 0. On the contrary, at a late time (Phase II), dust could reform and reddening will reappear, and the CI presents a blue-to-red change, which provides a decrease in the CI with ΔCI < 0. We note that Phase II might not occur, since the photodestruction could be irreversible (Perna & Lazzati 2002; Lazzati & Perna 2003; Perna et al. 2003).

The true magnitudes in the optical bands are estimated through considerations of extinctions (in both our Galaxy and the host galaxy) and the spectral k-correction. We display the distributions of all correction factors in Figure 19. The factor that affects the achromatic energy band is given by (−A^G–K–A^H). This quantity is distributed in [−4.29, 0.22], and, as seen in Figure 19(b), it is negligible compared to the brightness (Data Set III).

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The factor affecting the CIs can be characterized by the color excess, CE, such that

$$\begin{eqnarray}{\mathrm{CI}}_{{\rm{T}}} & = & {m}_{{\rm{O}}}({\lambda }_{1})-{A}^{{\rm{G}}}({\lambda }_{1})-K-{A}^{{\rm{H}}}({\lambda }_{1})\\ & & -\{{m}_{{\rm{O}}}({\lambda }_{2})-{A}^{{\rm{G}}}({\lambda }_{2})-K-{A}^{{\rm{H}}}({\lambda }_{2})\}\\ & = & {\mathrm{CI}}_{{\rm{O}}}-\{{A}^{{\rm{G}}}({\lambda }_{1})-{A}^{{\rm{G}}}({\lambda }_{2})+{A}^{{\rm{H}}}({\lambda }_{1})-{A}^{{\rm{H}}}({\lambda }_{2})\}\\ & = & {\mathrm{CI}}_{{\rm{O}}}-\mathrm{CE},\end{eqnarray} \tag{ 17 }$$

where A^H is the magnitude of the host dust extinction and A^G is the magnitude of our Galaxy dust extinction. Here the CE includes contributions from our Galaxy and the host galaxy, while the k-correction factor is the same for both bands. Therefore, CE = {A^G(λ₁) − A^G(λ₂) + A^H(λ₁) − A^H(λ₂)} is the total correction, ranging from 0.00 to 0.99. Figure 19(c) presents the CE as a function of CI. Since both the CI and CE have similar magnitudes, uncertainties on the corrections have large effects on the value of the CIs.

5.2.3. Effects of the Supernova Emission Component

5.2.4. Statistical Analysis

We perform a statistical analysis considering only variable CIs to explore possible physical origins. Figure 20(a) shows the distribution of variable CIs. If the observed variation is due to the crossing of the cooling spectral break, then either positive or negative values of ΔCIs are expected, depending on the density profile of the circumburst medium. As discussed above, a positive value of ΔCIs is expected in the early afterglow phase (earlier than 1 hr) owing to dust destruction. Variable CIs for the SN are defined as the identification of a GRB-SN bump at a later time, which is confirmed in published papers (see Section 5.2.3).

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Figure 20(b) presents the temporal evolution of the ratio of varying CIs to the total number of varying CIs in a given time interval. We group the ratios depending, in turn, on (i) the identification of the SN component and whether the ΔCIs are part of it and (ii) the sign of the ΔCIs.

Among the positive ΔCIs (64% of all cases, labeled II in Figure 20(a)) we find two peaks, at around 500 s and ∼10⁶ s. The early peak is consistent with being due to dust extinction, corresponding to 12% of the ΔCIs. These cases are marked by the red line in the figure. The peak at 10⁶ s is consistent with effects of ν_c crossing the observed band in a wind environment, causing a red-to-blue change, i.e., a shallowing of the spectrum (yellow line in the figure), including 43% of the variable CIs.

Likewise, among the negative ΔCIs (36% of all cases, labeled I in Figure 20(a)), there are two peaks, one at around 100 s and one at ∼10⁵–10⁶ s. The latter peak contains ΔCIs that are consistent with effects of ν_c crossing the observed band in an ISM environment, causing a blue-to-red change, i.e., a steepening of the spectrum (green line in the figure), corresponding to 30% of the cases. For the bursts that exhibit a color change at around 100 s (blue curve in the figure), the most natural explanation is the transition from reverse-shock (RS) emission to forward-shock (FS) emission. Early on, the emission is likely dominated by the RS component and the optical band is below ν_c,r (bluer spectrum). At around 100 s, there is likely a transition from the RS to the FS emission (Type II light curve reported in Zhang et al. 2003). At this time, the FS may be already above ν_c,f. This gives a redder color. For the wind model (Kobayashi & Zhang 2003; Wu et al. 2003), a similar situation is expected. This case includes ∼5% of ΔCIs.

Finally, the SN cases dominate the late-time variations (purple curve in the figure), the last one corresponding to 10% of the variable CIs.