Model-Independent Constraints on Lorentz Invariance Violation via the Cosmographic Approach

Since Lorentz invariance plays an important role in modern physics, it is of interest to test the possible Lorentz invariance violation (LIV). The time-lag (the arrival time delay between light curves in different energy bands) of Gamma-ray bursts (GRBs) has been extensively used to this end. However, to our best knowledge, one or more particular cosmological models were assumed {\it a priori} in (almost) all of the relevant works in the literature. So, this makes the results on LIV in those works model-dependent and hence not so robust in fact. In the present work, we try to avoid this problem by using a model-independent approach. We calculate the time delay induced by LIV with the cosmic expansion history given in terms of cosmography, without assuming any particular cosmological model. Then, we constrain the possible LIV with the observational data, and find weak hints for LIV.


I. INTRODUCTION
The rest of this paper is organized as follow. In Sec. II, we briefly review the formalism of the time delay ∆t induced by LIV. In Sec. III, we derive the cosmic expansion history in terms of cosmography with respect to redshift z at first. Then, we constrain the LIV parameters together with the cosmographic parameters by using the time delay data from GRBs, the observational data from type Ia supernovae (SNIa) and the baryon acoustic oscillation (BAO). Note that we adopt the Markov Chain Monte Carlo (MCMC) technique in doing this. As is well known, there exists a divergence problem in cosmography with respect to redshift z when z > 1. Thus, another cosmography with respect to the so-called y-shift y ≡ z/(1 + z) has been proposed in the literature, which alleviates the divergence problem since y < 1 in the range of 0 ≤ z < ∞. In Sec. IV, we obtain the observational constraints on LIV via the cosmographic approach with respect to y-shift. The brief concluding remarks are given in Sec. V.
Following the standard procedures given in e.g. [6] (calculating the comoving path in the expanding universe is the key), one can finally get the LIV-induced time delay between photons with energies E high and E low as (see e.g. [11,12,14]) where h(z) ≡ H(z)/H 0 is the dimensionless Hubble parameter, and z is the redshift of GRB. Following most of the relevant works in the literature (e.g. [5][6][7][8][9][10][11][12][13][14]16]), we only consider the case of n = 1, s ± = +1 and ξ 1 = 1 in the present work. In this case, Eq. (4) becomes Note that ∆t LIV in the original published version of [5] lacked the factor (1 + z) in the integration, and it has been corrected in the Erratum while the conclusion was modified accordingly.
For a cosmic transient source (e.g. GRB), the observed time delay between two different energy bands should include five terms [30,31], ∆t obs = ∆t LIV + ∆t int + ∆t spe + ∆t DM + ∆t gra , where ∆t LIV is the LIV-induced time delay as mentioned above. ∆t int is the intrinsic (astrophysical) time delay, which means that photons with high and low energies do not leave the source simultaneously. This term is difficult to predict because we have no good understanding on the physics of source evolution. As in the literature (e.g. [5,9,16]), one can write it as ∆t int = b (1 + z) while the cosmic expansion has been taken into account, and the constant parameter b characterizes our ignorance. ∆t spe represents the potential time delay due to special relativistic effects if photons have a non-zero rest mass. Since modern experiments have provided the upper limits for the photon rest mass as m ph < 10 −18 eV/c 2 [32], this term is negligible in fact [31]. ∆t DM is the time delay contribution from the dispersion by the line-of-sight free electron content, which is also negligible for GRB photons [30]. ∆t gra represents the effect of gravitational potential along the propagation path of photons if the Einstein's equivalence principle (EEP) is violated. This term can be dropped since EEP is preserved in our case. Substituting Eq. (5) and ∆t int = b (1 + z) into Eq. (6), and neglecting other terms, we obtain [5,7,9,11,12,16] ∆t where a LIV ≡ ∆E/(H 0 E QG ), and where In order to constrain the possible LIV, one should calculate the theoretical time delay induced by LIV, ∆t th = a LIV K + b (1 + z), and then confront it with the observed one, ∆t obs . From Eq. (10), it is easy to see that the cosmic expansion history (characterized by h(z)) should be given in advance. As mentioned in Sec. I, we will present it via the cosmographic approach which is model-independent, rather than assuming a particular cosmological model (e.g. ΛCDM) as in the literature. The only necessary assumption of cosmography is the cosmological principle, so that the spacetime metric is the one of the Friedmann-Robertson-Walker (FRW) universe, in terms of the comoving coordinates (t, r, θ, φ), where a is the scale factor. Motivated by the inflation paradigm and the observational results from e.g. Planck 2015 data [33], in this work we only consider a spatially flat FRW universe with k = 0. Introducing the so-called cosmographic parameters, namely the Hubble constant H 0 , the deceleration q 0 , the jerk j 0 , the snap s 0 , one can expand the scale factor a in terms of a Taylor series with respect to cosmic time t [19,20], One of the most important quantities in cosmology is the luminosity distance d L = (c/H 0 ) D L , where the dimensionless luminosity distance D L is defined by [19] We can also expand the dimensionless luminosity distance D L (equivalent to d L ) in terms of a Taylor series with respect to redshift z (see e.g. [19][20][21]29] for details), Differentiating Eq. (14), we get Substituting Eqs. (16) and (15) into Eq. (10), we obtain In this approach, one can calculate the theoretical time delay ∆t th = a LIV K + b (1 + z) by using Eq. (17), without assuming any particular cosmological model. The LIV parameters a LIV and b, together with the cosmographic parameters, will be determined by using the observational data. Note that one can also obtain h(z) or 1/h(z) by using Eq. (16) with D L given in Eq. (15). This is useful when one calculates other quantities (e.g. D V (z) used below).

B. Observational data
Ellis et al. [4] have developed the systematic analysis of statistical samples of GRBs at a range of different redshifts, and they have introduced techniques from signal processing such as wavelet analysis to identify and correlate genuine features in the intensities observed in different energy bands. Later, using these techniques, Ellis et al. [5] compiled a time delay dataset from 35 GRBs with known redshifts from z = 0.168 to z = 6.29. The numerical dataset can be found in Table 1 of [5]. To constrain the possible LIV, we perform the χ 2 statistics. The χ 2 from the time-lags of GRBs is given by where ∆t obs and σ i are the observed time delay and the corresponding uncertainty given in Table 1 of [5], and ∆t th = a LIV K + b (1 + z) is the theoretical time delay, while K is given in Eq. (17). Clearly, the cosmographic parameters (characterizing the cosmic expansion history) cannot be well constrained by using only the time delay data from GRBs. So, the other observational data are needed. Here, we consider the JLA (joint light-curve analysis) dataset [34] consisting of 740 SNIa obtained by the SDSS-II and SNLS collaborations. The theoretical distance modulus is defined by [19,34] where the luminosity distance d L = (c/H 0 ) D L , and D L is given in Eq. (15). On the other hand, in the JLA dataset, the observed distance modulus is given by [34] where m B corresponds to the observed peak magnitude in rest frame B band. α and β are both nuisance parameters. X 1 and C are the stretch measure and the color measure of SNIa, respectively. M is a nuisance parameter representing some combination of the absolute magnitude of a fiducial SNIa and the Hubble constant H 0 . The χ 2 from JLA SNIa reads [34] where ∆µ = µ obs − µ th , and C is the covariance matrix of µ. It is equivalent to [35] (see also e.g. [36]) where ∆m = m B − m mod , and while H 0 in d L can be absorbed into M. The numerical data of m B , X 1 , C, and the covariance matrix C can be found from the JLA dataset [34,37]. The nuisance parameters M, α and β could be marginalized (note that H 0 is also a nuisance parameter and it can be absorbed into M, and hence the JLA SNIa dataset is Hubble-free in fact). It is worth noting that since March 2014, the JLA likelihood plugin is included in the official release of the MCMC code CosmoMC [38,39]. We can further consider the observational data from BAO. Here we use the data D V (0.35)/D V (0.2) = 1.736 ± 0.065 from SDSS Collaboration [40], where D V (z) is related to the angular diameter distance d A and the luminosity distance d L or D L according to [40,41] in which we have used the well-known relation [19,22,24]) and Note that the factor c/H 0 will be cancelled in D V (0.35)/D V (0.2) and hence it is also Hubble-free. One can calculate D V (z) by using D L given in Eq. (15), from Eq. (16). The χ 2 from BAO is given by Note that actually there exist other BAO data in the literature, such as the observational data of A ≡ Ω [40,44,45]. However, they will introduce one or more free model parameters such as Ω m0 , Ω b , while the Hubble constant H 0 is no longer a nuisance parameter. This is a drawback, and makes the constraints loose. So, we do not consider such types of BAO data in this work. Similarly, we also do not consider the observational data from cosmic microwave background (CMB), since one or more free model parameters, e.g. Ω m0 , and/or Ω b , H 0 , should be introduced.  Table I for details.

C. Constraints on LIV
Here, we constrain the LIV parameters a LIV and b, together with the cosmographic parameters, by using the observational data mentioned in Sec. III B. Note that we use the MCMC code CosmoMC [38,39] in doing this. The cosmographic formulae for D L and K are given in Eqs. (15) and (17) respectively, while one can calculate D V (z) by using D L given in Eq. (15), and z/h( As mentioned above, the Hubble constant H 0 (which is also the first cosmographic parameter) has not been involved in the time delay data from GRBs and the D V (0.35)/D V (0.2) data from BAO, while it is a nuisance parameter in the JLA SNIa dataset and can be marginalized. On the other hand, if one consider the cosmographic formulae only up to 2nd order, the corresponding Taylor series cannot be enough general to include many cosmological models as its special cases, and hence the conclusions  Table I for details. become not so robust. However, if one consider the cosmographic formulae up to very high order, too many cosmographic parameters will be involved and hence the constraints become very loose. On balance, here we consider the cosmographic formulae up to 3rd and 4th orders one by one.
In the case of 3rd order cosmography with respect to redshift z (labeled as " z − j "), namely we only consider the Taylor series for D L (z), K(z), D V (z) ... up to 3rd order and ignore all the higher order terms O(z 4 ), there are only two free cosmographic parameters q 0 and j 0 besides the nuisance parameter H 0 . So, the free parameters under consideration are {q 0 , j 0 , a LIV , b}. The total χ 2 is given by By fitting the cosmographic formulae to the combined GRB+JLA+BAO observational data, we obtain the 1σ and 2σ constraints on the LIV parameters (a LIV , b) and the cosmographic parameters (q 0 , j 0 ), which are presented in the 2nd column of Table I. We also present the 2D marginalized 1σ and 2σ contours and the 1D distributions of the LIV parameters and the cosmographic parameters in Fig. 1. From Fig. 1 and the 2nd   with a non-zero a LIV (slightly beyond 1σ confidence region), while a LIV = 0 is still consistent with the observational data within 2σ confidence region.
In the case of 4th order cosmography with respect to redshift z (labeled as " z − s "), namely we only consider the Taylor series for D L (z), K(z), D V (z) ... up to 4th order and ignore all the higher order terms O(z 5 ), there are three free cosmographic parameters q 0 , j 0 and s 0 besides the nuisance parameter H 0 . So, the free parameters under consideration are {q 0 , j 0 , s 0 , a LIV , b}. By fitting the cosmographic formulae to the combined GRB+JLA+BAO observational data, we obtain the 1σ and 2σ constraints on the LIV parameters (a LIV , b) and the cosmographic parameters (q 0 , j 0 , s 0 ), which are presented in the 3rd column of Table I. We also present the 2D marginalized 1σ and 2σ contours and the 1D distributions of the LIV parameters and the cosmographic parameters in Fig. 2. From Fig. 2 and the 3rd column of Table I, we find again a weak hint for LIV with a non-zero a LIV (beyond 1σ confidence region), while a LIV = 0 is still consistent with the observational data within 2σ confidence region.
So, in both cases of cosmography with respect to redshift z, LIV with a non-zero a LIV is slightly favored by the observational data. On the other hand, from Figs. 1, 2, and Table I, it is easy to see that in both cases, the deceleration parameter q 0 is negative beyond 2σ confidence region, and the jerk j 0 is positive also beyond 2σ confidence region. Thus, from the definitions in Eq. (12), this means that an accelerating universe (q 0 < 0) is strongly favored, while the acceleration is still increasing (j 0 > 0).

A. Cosmographic approach with respect to y-shift
It is easy to see that the key of cosmography is to expand the quantities under consideration as a Taylor series. In the original version of cosmography, the relevant quantities are expanded with respect to redshift z. However, it is well known that such a Taylor series converges only for small z around 0, and it might diverge at high redshift z > 1. A possible remedy is to replace z with the so-called y-shift, y ≡ z/(1 + z) (see e.g. [22-24, 26, 29]). In this case, y < 1 holds in the whole cosmic past 0 ≤ z < ∞, and hence the Taylor series with respect to y-shift converges. So, here we also consider the cosmographic approach with respect to y-shift, y ≡ z/(1 + z).
We can expand the dimensionless luminosity distance D L (equivalent to d L ) in terms of a Taylor series with respect to y (see e.g. [21,23,29] for details),  The mean with 1σ and 2σ uncertainties of the LIV parameters and the cosmographic parameters for the cases of 3rd order (labeled as " y − j ", the 2nd column) and 4th order (labeled as " y − s ", the 3rd column) cosmography with respect to y-shift y = z/(1 + z), respectively. Note that aLIV and b are given in units of seconds, while q0, j0, s0 are all dimensionless. See the text for details.
Alternatively, one can derive Eq. (26) by substituting z = y/(1 − y) = y + y 2 + y 3 + y 4 + O(y 5 ) into Eq. (15) and then rearranging it as a series in terms of y. Similarly, we can also substitute z = y/(1−y) = y + y 2 + y 3 + y 4 + O(y 5 ) into Eq. (17) and then rearrange it as a series in terms of y, Of course, one can also expand D V as a Taylor series in terms of y by substituting z = y/(1 − y) = y + y 2 + y 3 + y 4 + O(y 5 ) into Eq. (24) and then rearranging it as a series in terms of y. Note that Eq. (16) is still useful in doing this.

B. Constraints on LIV
Again, we constrain the LIV parameters a LIV and b, together with the cosmographic parameters, by using the observational data mentioned in Sec. III B. This is similar to Sec. III C actually, but the cosmographic formulae for D L , K, D V ... should instead use the ones with respect to y-shift y = z/(1 + z) given in Sec. IV A.
In the case of 3rd order cosmography with respect to y-shift y = z/(1 + z) (labeled as " y − j "), namely we only consider the Taylor series for D L (y), K(y), D V (y) ... up to 3rd order and ignore all the higher order terms O(y 4 ), there are only two free cosmographic parameters q 0 and j 0 besides the nuisance parameter H 0 . So, the free parameters under consideration are {q 0 , j 0 , a LIV , b}. The total χ 2 is given by By fitting the cosmographic formulae to the combined GRB+JLA+BAO observational data, we obtain the 1σ and 2σ constraints on the LIV parameters (a LIV , b) and the cosmographic parameters (q 0 , j 0 ), which are presented in the 2nd column of Table II. We also present the 2D marginalized 1σ and 2σ contours and the 1D distributions of the LIV parameters and the cosmographic parameters in Fig. 3. From Fig. 3 and the 2nd column of Table II, we find a notable hint for LIV with a non-zero a LIV (around 2σ confidence region).
In the case of 4th order cosmography with respect to y-shift y = z/(1 + z) (labeled as " y − s "), namely we only consider the Taylor series for D L (y), K(y), D V (y) ... up to 4th order and ignore all the higher order terms O(y 5 ), there are three free cosmographic parameters q 0 , j 0 and s 0 besides the nuisance parameter H 0 . So, the free parameters under consideration are {q 0 , j 0 , s 0 , a LIV , b}. By fitting the cosmographic formulae to the combined GRB+JLA+BAO observational data, we obtain the 1σ and 2σ constraints on the LIV parameters (a LIV , b) and the cosmographic parameters (q 0 , j 0 , s 0 ), which are presented in the 3rd column of Table II. We also present the 2D marginalized 1σ and 2σ contours and the 1D distributions of the LIV parameters and the cosmographic parameters in Fig. 4. From Fig. 4 and  the 3rd column of Table II, we find that a LIV = 0 is fully consistent with the observational data, namely there is no evidence for LIV.
Although the results about LIV are quite different in both the cases of cosmography with respect to y-shift y = z/(1 + z), from Figs. 3, 4, and Table II, it is easy to see that in both cases, the deceleration parameter q 0 is negative beyond 2σ confidence region. Thus, from the definition in Eq. (12), this means that an accelerating universe (q 0 < 0) is strongly favored.  been extensively used to this end. However, to our best knowledge, one or more particular cosmological models were assumed a priori in (almost) all of the relevant works in the literature. So, this makes the results on LIV in those works model-dependent and hence not so robust in fact. In the present work, we try to avoid this problem by using a model-independent approach. We calculate the time delay induced by LIV with the cosmic expansion history given in terms of cosmography, without assuming any particular cosmological model. Then, we constrain the possible LIV with the observational data from GRBs, SNIa and BAO, and find weak hints for LIV with non-zero a LIV in 3 of 4 cases of cosmography considered in the present work. It is of interest to compare the 4 cases of cosmography considered here. As mentioned above, they are labeled as " z − j ", " z − s ", " y − j " and " y − s ", respectively. Since they have different free parameters and the correlations between model parameters are fairly different, it is not suitable to directly compare their confidence level contours. Instead, it is more appropriate to compare them from the viewpoint of  Comparing the four cases of cosmography considered in the present work, namely 3rd order (labeled as " z − j ") and 4th order (labeled as " z − s ") cosmography with respect to redshift z, as well as 3rd order (labeled as " y − j ") and 4th order (labeled as " y − s ") cosmography with respect to y-shift y = z/(1 + z). See the text for details.
goodness-of-fit. A conventional criterion for model comparison in the literature is χ 2 min /dof , in which the degree of freedom dof = N − κ, while N and κ are the number of data points and the number of free model parameters, respectively. On the other hand, there are other criteria for model comparison in the literature. The most sophisticated criterion is the Bayesian evidence (see e.g. [46] and references therein). However, the computation of Bayesian evidence usually consumes a large amount of time and power. As an alternative, one can consider some approximations of Bayesian evidence, such as the so-called Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC). The BIC is defined by [47] where L max is the maximum likelihood. In the Gaussian cases, χ 2 min = −2 ln L max . So, the difference in BIC between two models is given by ∆BIC = ∆χ 2 min + ∆κ ln N . The AIC is defined by [48] AIC = −2 ln L max + 2κ .
The difference in AIC between two models is given by ∆AIC = ∆χ 2 min + 2∆κ. In Table III, we present χ 2 min /dof , ∆BIC and ∆AIC for the 4 cases of cosmography considered in this work. Note that " z − s " has been chosen to be the fiducial model when we calculate ∆BIC and ∆AIC. Clearly, from the viewpoint of all the three criteria χ 2 min /dof , BIC and AIC, " z − s " is the best, and " y − s " is the worst. Together with the fact that we find no evidence for LIV only in the case " y − s " while there are weak hints for LIV in the other three cases " z − s ", " y − j " and " z − j ", we conclude that LIV with non-zero a LIV is slightly favored by the observational data.
In the literature, there exist many relevant works on the possible LIV. In addition to the weak evidence for LIV found in [5] (its Erratum should be seriously considered), some further works (e.g. [9,11,12,15]) supported this conclusion of [5]. In particular, a strong evidence for LIV was claimed in [11,12]. Our results obtained in the present work could be regarded as a new support. On the other hand, one should also be aware of the contrary claim that there is no evidence for LIV (see e.g. [16]). Besides, most of the relevant works in the literature kept silence and just put a lower bound on the possible LIV energy scale E QG . Therefore, the debate on LIV has not been settled so far. More and better observational data from e.g. GRBs are needed. New ideas to test LIV are also desirable.
In the present work, we consider the cosmographic approaches with respect to redshift z and y-shift y = z/(1 + z). The first one might diverge at high redshift z > 1, while the second one can alleviate this problem since y < 1 in the whole cosmic past 0 ≤ z < ∞, and hence the Taylor series with respect to y converges. However, there still exist several serious problems in the case of y = z/(1 + z). The first is that the error of a Taylor approximation throwing away the higher order terms will become unacceptably large when y is close to 1 (say, when z > 9). The second is that the cosmography in terms of y = z/(1 + z) cannot work well in the cosmic future −1 < z < 0. The Taylor series with respect to y = z/(1 + z) does not converge when y < −1 (namely z < −1/2), and it drastically diverges when z → −1 (it is easy to see that y → −∞ in this case). So, the y-shift cosmography fails to predict the future evolution of the universe. In [29], two new generalizations of cosmography inspired by the Padé approximant have been proposed, which can avoid or at least alleviate the problems of ordinary cosmography mentioned above.
The model-independent constraints on LIV via these two new cosmographic approaches proposed in [29] deserve further investigation.
In most of the relevant works on LIV (including the present work), the intrinsic time-lag of GRBs is actually oversimplified by assuming ∆t int = b (1 + z). Noting that the factor (1+z) comes from the cosmic expansion, this assumption means that all GRBs have the same intrinsic time delay (characterized by the constant parameter b) in the source frame. In e.g. [15], an energy-dependent time-lag was proposed. We consider that a more realistic assumption for the intrinsic time-lag of GRBs is important to robustly constrain the possible LIV in the future work. Deeper understanding on the observed time-lag of GRBs is also desirable (nb. Eq. (6)), especially the ones other than LIV-induced time delay.
As mentioned in Sec. I, the possible LIV is commonly accompanied with a non-trivial dispersion relation. A signal of energy E that travels a distance L acquires a time delay (measured with respect to the ordinary case of an energy-independent speed c), namely ∆t ∼ ξ(E/E QG )(L/c). Although the QG effect is expected to be very weak, a very long distance L can still make it testable. This point can be clearly seen from Eq. (5), namely a large time delay ∆t LIV follows a high redshift z. GRBs are among the most powerful sources in the universe. Their high energy photons in the gamma-ray band are almost immune to dust extinction, and hence they have been observed up to redshift z ∼ 8 − 9 [51,52], while the maximum redshift of GRBs is expected to be 10 or even larger [53,54]. Therefore, GRBs at high redshift can be used to test the possible LIV which induces a large time delay. In the present work, we find weak hints for LIV by using the time delay dataset from 35 GRBs with redshifts up to z = 6.29 [5] via the cosmographic approach. Our results suggest that the possible LIV should be taken seriously. The physical mechanism for LIV might be the spacetime foam predicted in most of the quantum gravity theories (e.g. string theory, loop quantum gravity, and doubly special relativity). In addition, we would like to mention the so-called Standard-Model Extension (SME) [49], which also provides a field theory framework for LIV. The deep physics behind LIV deserves serious consideration.