Lorentz invariance violation and simultaneous emission of electromagnetic and gravitational waves 1
E. Passos,∗ 1 M. A. Anacleto,†
1,2
F. A. Brito,‡ 4 O. Holanda,§ 1 G. B. Souza,¶ and 3 C. A. D. Zarro∗∗
1
arXiv:1612.02769v1 [hep-th] 8 Dec 2016
Departamento de F´ısica, Universidade Federal de Campina Grande, Caixa Postal 10071, 58429-900, Campina Grande, Para´ıba, Brazil. 2 Departamento de F´ısica, Universidade Federal da Para´ıba, Caixa Postal 5008, Jo˜ ao Pessoa, Para´ıba, Brazil. 3 Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 21945, Rio de Janeiro, Rio de Janeiro, Brazil. and 4 Departamento de F´ısica Te´ orica, Instituto de F´ısica, Universidade do Estado do Rio de Janeiro, Rua S˜ ao Francisco Xavier 524, 20550-013, Maracan˜ a, Rio de Janeiro, Brazil In this work, we compute some phenomenological bounds for the electromagnetic and massive gravitational high-derivative extensions supposing that it is possible to have an astrophysical process that generates simultaneously gravitational and eletromagnetic waves. We present a Lorentzviolating (LIV) higher-order derivative, following the Myers-Pospelov approach, to electrodynamics and massive gravitational waves. We compute the corrected equation of motion of these models, their dispersion relations and the velocities. The LIV parameters for the gravitational and electromagnetic sector, ξg and ξγ respectively were also obtained for three different approaches: luminal photons, time delay of filght and the difference of graviton and photon velocities. These LIV parameters depend on the mass scales where the LIV-terms become relevant, M for the electromagnetic sector and M1 for the gravitational one. We obtain, using the values for M and M1 found in the literature, that ξg ∼ 10−2 , which is expected to be phenomenolocally relevant and ξγ ∼ 103 , which cannot be suitable for an effective LIV theory. However, we show that ξγ can be interesting in a phenomological point of view if M ≫ M1 . Finally the difference between the velocities of the photon and the graviton was calculated and our result, vγ − vg . 0.49 × 10−19 , is compatible with the results already presented in the literature.
I.
INTRODUCTION
The Lorentz invariance violating (LIV) theories have been extensively studied at high energy systems. The main focus is to develop an effective probe to test the phenomenological limits of Lorentz invariance as a direct consequence on the Planck scale physics such as the fuzzy nature of spacetime provided by quantum gravity theories. In this context, the possible effects related to LIV are given by energy and helicity dependent photon propagation velocities. The LIV parameters bounds on energy can be inferred by measuring the arrival times of photons with different energies emmited almost simultanously from distant objects [1]. In order to measure such bounds, it is necessary an ultra high energy phenomena such as a a gamma-ray burst (GRB) [2–5] or a flare of an active galactic nucleus [6, 7]. The LIV parameters can also be constrained by measuring how the polarization direction of an x-ray beam of cosmological origin changes as function of energy [8]. Such observations have been used as astrophysical laboratories to verify the possible occurrences of LIV in nature [9–12]. One approach to investigate LIV effective theories was initially proposed by Myers- Pospelov by breaking the Lorentz symmetry after introducing mass operators of dimension-five along with a nondynamical four-vector nµ interacting with scalars, fermions, and photons fields [13]. If we restrict our attention only to photon sector, there is a single contribution of dimension-five operators which gives a correction of order ξγ p3 /MPl . The extension to dimension-n operators satisfies all the Myers-Pospelov approach criteria: (i) quadratic in the fields, (ii) one more derivative than the usual terms, (iii) gauge invariant, (iv) Lorentz invariant, except for the presence of an external four-vector nµ , (v) not reducible to lower dimensional operators at the equations of motion, and (vi) not reducible to a total derivative. In this set-up, one finds that for dimension-n operators, the correction is given by ξγ pn /(MPl )n−2 .
∗ Electronic
address:
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2 The detection of gravitational waves, reported by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo Collaborations [15, 16], opens a new window in observational cosmology and astrophysics. Particularly in astrophysics, the hitherto detected gravitational waves come from the merger of two black-holes. However, it is expected to also measure gravitational waves emmited during the merger of other compact astrophysics objects such as neutron stars or a black-hole and a neutron star. The merger of such two compact objects is supposed to be a very complex phenomena, probably involving electromagnetic waves or neutrino emission, hence to obtain new insights of the merger process, one can observe simultaneously the emission of gravitational waves and electromagnetic waves or neutrinos. It was reported, in the event GW150914 the observation of a gravitational wave [15] and a short gamma-ray burst (also detected in the same event GW150914 [17]) by the Fermi Gamma-Ray Space Telescope have been used to obtain constraints on LIV parameters [18–21] (see also [22]). This issue started an intense debate in the literature as some authors describe that this electromagnetic counterpart is not possible [23] and others showed its plausibility [24, 25]. The null results of simultaneous GRB emission and the other detected gravitational wave event [26, 27], GW151226 [16], apparently favours that the this simultaneous emission was unlikely, however it is not possible to conclude anything, as it there are only two detected gravitational waves events and the physical processes involved in eletromagnetic waves and neutrino emissions are not enterely understood. A transient GRB signal above 50 keV after 0,4 s after the detection of GW150914 was reported in Ref. [17]. This observational fact and a bound for the graviton mass, mg ≤ 10−22 eV, was used in Refs. [18–21] to obtain bounds constraining the difference between light and graviton speed and the energy scale where the LIV effects appear. Our goal is to extend these references by introducing high-order derivative operators which explicitly break Lorentz symmetry. The main purpose of this work is to consider the electromagnetic and gravitational dispersion relations produced by presence of the higher derivative operator in the effective actions. We aim to find new phenomenological constraints on LIV by using simultaneous measures of the gamma-ray busts (GRB) and gravitational wave (GW) produced by the same source, i.e, assuming that such signals emerge from black holes merger. The outline of this paper is as follows. In Sec. II, we have demonstrated that both the electromagnetic and linearized gravitational higher derivative extensions appear as terms of a power series associate to CPT-odd effective actions. In Sec. III, we use a dimension-five operator as modification to Maxwell Lagrangian. The associate dispersion relations are obtained. In Sec. IV, we also use a dimension-five operator to modify the massive Fierz-Pauli Lagrangian. The associated dispersion relations are obtained for massive and massless cases. In Sec. V, we discussed some phenomenological constraints. In Sec.VI, we present our conclusions. II.
THE HIGHER DERIVATIVE LIV EXTENSIONS A.
The electromagnetic sector
We consider a CPT-odd pure-photon action proposed by Carrol-Field-Jackiw (CFJ) [28] and through it we get higher derivative contributions of a power series. The CFJ action rewritten as Z M ξγ Sγ = − d4 x f µλν Fλµ Aν , f µλν ≡ εαµλν nα 4 Z M ξγ d4 x Aµ Πµν Aν (1) = 2 where ξγ is a dimensionless parameter, M is the mass scale where LIV effects emerges and Πµν = f µλν ∂λ is the electromagnetic LIV operator that enjoys the following properties: ∂µ Πµν = 0, nµ Πµν = 0, Πµν Πνβ = − δµβ (n · ∂)2 − n2 ∂ 2 − nβ ∂µ (n · ∂) − nµ ∂ 2 − ∂ β nµ (n · ∂) − n2 ∂µ and Πµν Πµν = 2 (n · ∂)2 − n2 ∂ 2 . Notice that the above effective action is gauge invariant (up to a surface term) under gauge transformations δAµ = ∂µ Λ. We now extend the CFJ action by replacing the quantity ξγ M Πµν to a power series such as X
l=1,3,...
ξγ ξγl ˆ + ... (Πµν )l = ξγ1 M Πµν + 3 Πµα Παβ Πβν = Πµν D l−2 (M ) M = ξγ1 M Πµν +
ξγ3 µν ˆ Π D + ... M
ˆ = (n · ∂)2 − n2 ∂ 2 is a LIV derivative operator. Inserting the series (2) into action (1) we get where D Z i h 1 ξγ ˆ ˆ λµ Aν + ... Sγ → Sγ = − d4 x M ξγ1 f µλν Fλµ Aν + 3 f µλν DF 2 M
(2)
(3)
3 We can rewrite the Eq. (3) in the form Z i h ξγ ξγ 1 ˆ d4 x M ξγ1 f µλν Fλµ Aν + 3 f µλρ nν nρ (∂λ Fµν )Fρσ − 3 f µλρ n2 η νρ (∂λ Fµν )Fρσ + ... . Sγ → Sγ = − 2 M M
(4)
Notice that we obtain dimension-five operators as extra terms of power series expansion in Eq.(3) (or Eq.(4)) which lead to cubic modifications of the dispersion relations of electromagnetic waves. These contributions obey the main Myers-Pospelov criteria [13]. B.
The gravitational sector
In analogy to the above electromagnetic case, we consider the following LIV extension to Fierz-Pauli action proposed in Ref. [29]: Z (M1 )3 ξg Sg = − d4 x f µλν hρµ ∂λ hρν , f µλν ≡ εαµλν nα (5) 2 where ξg is a dimensionless parameter, M1 is the mass scale where LIV in the gravitational sector become pronounced. Here the hµν is a second rank symmetric tensor characterizing weak metric fluctuations (hµν = gµν − ηµν , where gµν is the metric tensor of the curved space, ηµν = diag(−1, +1, +1, +1) is the metric tensor of the flat space and h = η µν hµν is the trace of hµν [30]). Notice that under gauge transformations δhµν = ∂µ ξν + ∂ν ξµ for a spacetime dependent gauge parameter ξµ (x) the action (5) implies in the following variation: δLg ∼ f µλν ξµ ∂λ ∂ρ hρν which does not vanish in general, so that the action Sg is not gauge invariant. This issue can be investigated using the Stuckelberg formalism to a massive spin-two field [30]. To the LIV case, see also Ref. [29]. Following last section, we rewrite the Eq. (5) in terms of a power series such as in the electromagnetic case. Again we replace the LIV operator to the following power series ξgl ξg ˆ + ... (Πµν )l = ξg1 M1 Πµν + 3 Πµν D (M1 )l−2 M1
X
l=1,3,...
(6)
such that 2
(M1 ) Sg → Sˆg = − 2
Z
i h ξg ˆ hρ + ... . d4 x M1 ξg1 f µλν hρµ ∂λ hρν + 3 f µλν hρµ ∂λ D ν M1
(7)
Therefore, we also derive a higher-dimension operators as extra terms of power series expansion which lead to cubic modifications of the dispersion relations of gravitational waves. Although the restriction on the gauge invariance, the extra contribution in Eq.(7) satisfies all the Myers-Pospelov criteria to construct LIV higher derivative operators. III.
THE EXTENDED ELECTRODYNAMICS
In this section we consider the second term of the Eq.(4) to modified classical electrodynamics (electromagnetic Maxwell-Myers-Pospelov model). In this case we analyze the dispersion relation of electromagnetic waves. A.
The model
Let us now derive the dynamics associated with the following Lagrangian 1 ξγ µλρ ν σ ξγ µλρ 2 νσ Lγ = − Fµν F µν − f n n (∂λ Fµν )Fρσ + f n η (∂λ Fµν )Fρσ 4 M M
(8)
where ξγ3 ≡ ξγ . Using the axial gauge nµ Aµ = 0, the equation of motion reads ∂ 2 η µν −
2ξγ µλν ˆ Aν = 0. f ∂λ D M
(9)
4 After a straightforward algebra we find that the free continuous spectrum associated with the equation of motion, Eq. (9), is governed by the following covariant dispersion relation: (kγ2 )2 − (2ξγ /M )2 (n · kγ )2 − n2 kγ2 which was also derived in Ref. [31]. B.
3
=0
(10)
Modified propagations to electromagnetic waves
The solutions to the above dispersion relation for the isotropic configuration, i.e. when nµ ≡ (1, ~0) is chosen to be purely time-like are investigated in Ref. [31]. From this isotropic configuration we generalize the dispersion relations associated with dimension−n operators: Eγ2 − kγ2 − 2λξγ(n)
kγn = 0, M n−2
kγ ≡ |~kγ |
(11)
with the two polarizations λ = ±1. For n = 3 we recover the cubic modifications reported in [13]. And, for n = 4, 5, ... we find new expressions due to the increase of the dimension of the LIV operator. Solving Eq. (11) for the energy, we obtain the frequency solutions q n−2 (n) . (12) Eγ = kγ 1 + 2λξγ kγ /M The dispersion relation (12) leads to a modified speed of light for a photon with momentum kγ : n−2 (n) 1 + nλξγ kγ /M ∂Eγ vγ ≡ =q n−2 . ∂kγ (n) 1 + 2λξγ kγ /M
(13)
This dispersion relation leads to rotations of the polarization of linearly polarized photons during their propagation (see, e.g., Refs. [11] and [32]) . Now we will carry out an expansion of the above expression for (kγ )n−2 ≪ 1/ 2ξγ (M )2−n . This leads to
vγ ≈ 1 + λ(n − 1)ξγ(n)
kγ M
n−2
.
(14)
Notice that for n ≥ 3 the speed vγ (λ=−) can exceed the speed of light introducing problems of causality (see also Ref. [31]). IV.
THE EXTENDED LINEARIZED GRAVITY A.
The Model
In this case we consider the second contribution in Eq.(7) as a modification in the dynamics of the massive FierzPauli action. Here we also analyze the dynamics associated with the dispersion relation of the gravitational waves. Considering the following Lagrangian (in our case, for brevity, ξg3 ≡ ξg ): Lg =
(M1 )2 h 1 1 ∂λ hµν ∂ λ hµν + ∂µ hνλ ∂ ν hµλ − ∂µ hµν ∂ν h + ∂λ h∂ λ h + 2 2 2 i ξg µλν 1 2 ρ µν 2 ˆ f hρµ ∂λ D h ν . m hµν h − h − 2 g M1
(15)
We will take this Lagrangian since it can describe a massive spin-two LIV theory. Notice that there is no gauge symmetry due to a mass term and the −1 coefficient in hµν hµν − h2 is dubbed as Fierz-Pauli tuning. In this manuscript, we are interested in the phenomenological aspects associated to Eq. (15).
5 The equations of motion from (15) are given as ✷hµν − ∂λ ∂ µ hλν − ∂λ ∂ ν hλµ + η µν ∂λ ∂σ hλσ + ∂ µ ∂ ν h 2ξg µλβ ˆ η αν hαβ = 0. f ∂λ D −η µν ✷h + m2g hµν − η µν h − M1
(16)
2ξg µλβ ˆ αν f ∂λ D η hαβ = 0. M1
(17)
Assuming that mg 6= 0, after applying ∂µ on Eq. (16) one obtains ∂µ hµν = ∂ ν h and plugging this back into the above equations of motion, we get ✷hµν − ∂ µ ∂ ν h + m2g (hµν − η µν h) −
Taking the trace of this equation, we find h = 0, which in turn implies that ∂µ hµν = 0. Provided that ∂µ hµν = 0 and h = 0, Eq. (17) reads h 2ξg µλβ ˆ αν i ✷ + m2g η αµ η βν − hαβ = 0, (18a) f ∂λ D η M1
After a straightforward algebra we find that the free continuous spectrum associated with Eq. (18a) is given by the following dispersion relation 3 2 2 (19) kg2 − m2g − 2ξg /M1 (n · kg )2 − n2 kg2 = 0.
Notice that for mg = 0, the Eq.(19) is equivalent to the Eq.(10), i.e., the electromagnetic case, given e.g. in Ref. [31]. B.
Modified propagations to gravitational waves
Moreover, we study the solutions for the dispersion relation given by Eq. (19) in the isotropic configuration, that is, for nµ = (1, ~0) chosen to be purely time-like, for dimension−n operators. Thus, we have Eg2 − kg2 − m2g − 2λξg(n)
kgn M1n−2
= 0,
kg ≡ |~kg |
with the two polarizations λ = ±1. Solving the Eq.(20) for Eg we find the frequency solutions q n−2 (n) Eg = kg2 1 + 2λξg kg /M1 + m2g .
(20)
(21)
Notice also that the solutions correctly reproduce the usual ones in the limit ξg → 0 given in Ref. Hinterbichler:2011tt. We assume here the graviton velocity, vg , is given by the group velocity determined from the dispersion relation (21), that is n−2 (n) 1 + nλξg kg /M1 ∂Eg vg ≡ (22) = q n−2 ∂kg (n) + (mg /kg )2 1 + 2λξg kg /M1
Now expanding for large momenta kg2 ≫ m2g , but keeping (kg )n−2 ≪ 1/ 2ξg (M1 )2−n as before, we find, n−2 n−2 2 kg 1 mg kg (n) (n) 1 + λnξg vg ≈ 1 + λ(n − 1)ξg − . M1 2 M1 kg
(23)
In the limit M1 ≫ mg (for n ≥ 3), the Eq.(23) takes the form n−2 m2g kg (n) vg ≈ 1 − 2 + λ(n − 1)ξg . (24) 2kg M1 n−2 (n) (n) , then the graviton travels slower than By considering vg(−) , notice that if ξg > 0, with m2g /kg2 > |ξg | kg /M1 n−2 (n) (n) 2 2 , then the graviton would propagate faster light speed. On the other hand, if ξg < 0 and mg /kg < |ξg | kg /M1 than light speed. To massless gravitons (mg = 0) we have, n−2 kg (n) . (25) vg ≈ 1 + λ(n − 1)ξg M1
6 which is similar to Eq. (14), i.e., the group velocity of photons. In the absence of LIV regimes, i.e., (ξg = 0), we get vg ≈ 1 −
m2g 2kg2
(26)
as in the usual case [19]. V.
PHENOMENOLOGICAL ASPECTS
In the following we consider the Eq.(14) for photons and Eq.(24) for massive gravitons to impose the upper bounds for the ξg , ξγ − LIV. To do this, we use the Fermi Gamma-Ray Burst Monitor (GMB)-LIGO observations associated with a transient source based in the following measured of time arrival delay: ∆t ∼ 0.40 s between the gamma-ray burst and the gravitational wave [17]. A.
Approach one: luminal photons
In this approach we consider, following Ref. [19], that ∆vg(−) = ∆vγ (−) ξ
γ =0
(for luminal photons case) so that
(n) m2g ξg 1 = (M1 )n−2 2(n − 1) (kg )n
(27)
where we have considered kg ∼ 100 Hz ∼ 4.13 × 10−13 eV and mg . 10−22 eV as the energy and the mass of the graviton estimated by LIGO [15]. By inserting these values into Eq.(27), one finds ξg(n)
(10)13n−44 . 2(n − 1) (4.13)n
M1 eV
n−2
.
(28)
The above expression can impose an upper bound for the ξg − LIV parameter. Notice that for n = 3, we get ξg(n=3) . (3.54 M1 ) × 10−8 (eV)−1
(29)
corresponds to a mass-scale dependent parameter. Particularly, if we use M1 ∼ 105 eV [19], we obtain that this upper bound is ξg ∼ 10−2 , which can be relevant phenomenologically. B.
Approach two: time delay of flight
The difference ∆t = ∆tg − ∆tγ between the propagation of the gravitational and electromagnetic waves is given by [? ] ∆t = ∆ta − (1 + z)∆te
(30)
where ∆ta (measured quantity) is the arrival delay observed at the Earth and ∆te (unknown quantity) is the emission delay at the source with redshift z. Here we assumethat ∆te = 0 (the simultaneous emission of gravitational and electromagnetic waves) to derive constraints on LIV by velocities of the gravitons and photons. Thus for spatially flat Universe, Ωk = 0 we have for luminal gravitons Z z 1 1 dz ′ p ∆ta = ∆tg − ∆tγ = H0−1 . (31) − vγ (−) vg ξg =0 Ωm (1 + z ′ )3 + ΩΛ 0
where H0 = 67.8 Km (s Mpc)−1 is the Hubble constant (H0−1 = 4.55 × 1017 s) with Ωm and ΩΛ being the matter and dark energy density parameters, respectively. Inserting Eqs.(14) and (25) into Eq.(31), we find 2 n−2 Z z mg dz ′ (n) kγ p . (32) − ξ ∆ta = (n − 1)H0−1 γ 2kg2 M Ωm (1 + z ′ )3 + ΩΛ 0
7 Thus, from Eq.(32) we get " Z z −1 # (n) m2g dz ′ ξγ ∆ta 2−n p = −(kγ ) − (M )n−2 2kg2 (n − 1)H0−1 Ωm (1 + z ′ )3 + ΩΛ 0
(33)
Now solving the integral for Ωm = 0.31, ΩΛ = 0.69 at a redshift z = 0.09 and using the previously assumed values for ∆ta = 0,4 s, kg ∼ 4.13 × 10−13 eV and kγ > 50keV being the photon energies for transient source measured by GMB [17], we obtain (0.58)n − 1.57 × 10−(4n+11) M n−2 (n) ξγ . (34) (n − 1)(5.0)n−2 eV Notice that for n = 3, we have ξγ(n=3) . (1.70 M ) × 10−25 (eV)−1
(35)
which also corresponds to a mass-scale dependent parameter. In particular, inserting M ∼ 1028 eV, as suggested, (n=3) in Refs. [7, 19] into Eq. (35), ξγ ∼ 103 is gotten. Hence for this very M value, our result is not suitable for any realistic phenomenology. One then concludes that eiher the value of M has to be modified (see also Ref. Passos:2016bbc, where another energy scale, namely the Hoˇrava-Lifshitz one is also introduced to give more realistic bounds) or this term cannot be present in the description of a LIV effective theory. To complete our phenomenological analysis let us compare Eq.(28) with Eq.(34). As a consequence, we find the following relationship n−2 (n) ξg M1 (5.0)n−2 × 1017n−33 . (36) = (n) (0.58)n − 1.57 (4.13)n M ξγ Therefore, for (n = 3), we find,
(n=3)
ξg
(n=3)
ξγ
= 4.17 × 1017
M1 . M
(37)
Notice that if M ≫ M1 , the above quantity may lead to a realistic constraint. C.
Approach three: Difference between the light and graviton velocities
In this approach we use the above results to ξg and ξγ parameters to constrain the difference between the velocities of electromagnetic and gravitational waves. First, we insert the Eq.(28) into Eq.(24) to find (for λ = −1) n−2 m2g 10(13n−44) kg vg = 1 − 2 − . (38) 2kg 2(4.13)n eV Also using the previous values for mg and kg we obtain vg . 1 − 0, 59 × 10−19 . Now inserting the Eq.(34) into Eq.(14) we find (also at λ = −1) (0.58)n − 1.57 × 10−(4n+11) kγ n−2 vγ = 1 − (5.0)n−2 eV
(39)
(40)
and by using the previous value for kγ , we have for n = 3 vγ . 1 − 1, 07 × 10−19 .
(41)
Therefore the difference between velocities given by Eq.(39) and Eq.(41) is given as vγ − vg . 0.49 × 10−19 which is the approximately the bounds found in [19] — see also [20].
(42)
8 VI.
CONCLUSIONS
In this work, we analyze the LIV effects from electromagnetic and gravitational higher derivative operators using the Myers-Pospelov approach to obtain LIV effective theories. First, we extend the electromagnetic and massive gravitational actions to include LIV high order derivate terms. Then we compute the equations of motion, the dispersion relations for these sectors and the photon and graviton velocities. Assuming that the same process that generated the detected the gravitational waves also emits eletromagnetic waves, also detected by other means, bounds for the LIV parameters for electromagnetic, ξγ , and massive gravitational, ξg , sectors are obtained for three approaches, namely, luminal photons, time delay of flight and the difference between photon and graviton velocities. For both approaches, there is a dependence of ξg and ξγ on the respective mass scales M1 and M , where the LIV effects become relevant. Using the value for M1 obtained in Ref. [19], it is gotten that ξγ ∼ 10−2 , and this is expected to be phenomenological relevant. For the time delay of flight approach and the value of M given in Refs. [7, 19], it is found that ξg ∼ 103 , which cannot represent any realistic LIV scenario, however, the ratio between ξg and ξγ , Eq. (37), can be made phenomenological relevant if M ≫ M1 , which is satisfied, even if we consider that M has to be changed to make ξγ ∼ 1 in Eq. (35). Finally for the difference of photon and graviton velocities, we found compatible bounds to the ones already obtained in the literature. Acknowledgments
We would like to thank to CNPq for partial financial support. C. A. D. Z. is thankful for the kindness and hospilaty of the Physics Department at Federal University of Campina Grande, where part of this work was carried out.
9
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