Deformed Relativistic Symmetries in FRW spacetime Giacomo Rosati - - PowerPoint PPT Presentation
Deformed Relativistic Symmetries in FRW spacetime Giacomo Rosati Institute for Theoretical Physics, University of Wroc law Experimental search for quantum gravity Trieste September 4, 2014 G.Amelino-Camelia+A.Marcian
Institute for Theoretical Physics, University of Wroc law
September 4, 2014
G.Amelino-Camelia+A.Marcian´
G.Amelino-Camelia+G.R.,arXiv:XXXX.XXXX,forthcoming
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation)
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) DSR (Deformed Relativistic Symmetries)
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV DSR (Deformed Relativistic Symmetries)
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV Coordinates of different (inertial) observers are connected by ordinary Poincar´ e transformations DSR (Deformed Relativistic Symmetries)
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV Coordinates of different (inertial) observers are connected by ordinary Poincar´ e transformations E2 − p2+λLIV Ep2 = m2 holds only for a preferred observer → Poincar´ e symmetries are broken DSR (Deformed Relativistic Symmetries)
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV Coordinates of different (inertial) observers are connected by ordinary Poincar´ e transformations E2 − p2+λLIV Ep2 = m2 holds only for a preferred observer → Poincar´ e symmetries are broken Different (inertial) observers, by the same measuring procedure, obtain different values for λLIV DSR (Deformed Relativistic Symmetries)
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV Coordinates of different (inertial) observers are connected by ordinary Poincar´ e transformations E2 − p2+λLIV Ep2 = m2 holds only for a preferred observer → Poincar´ e symmetries are broken Different (inertial) observers, by the same measuring procedure, obtain different values for λLIV DSR (Deformed Relativistic Symmetries) The laws of motion are modified at a scale ℓDSR
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV Coordinates of different (inertial) observers are connected by ordinary Poincar´ e transformations E2 − p2+λLIV Ep2 = m2 holds only for a preferred observer → Poincar´ e symmetries are broken Different (inertial) observers, by the same measuring procedure, obtain different values for λLIV DSR (Deformed Relativistic Symmetries) The laws of motion are modified at a scale ℓDSR Different (inertial) observers, by the same measuring procedure, obtain the same value for ℓDSR (ℓDSR is a relativistic invariant as c)
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV Coordinates of different (inertial) observers are connected by ordinary Poincar´ e transformations E2 − p2+λLIV Ep2 = m2 holds only for a preferred observer → Poincar´ e symmetries are broken Different (inertial) observers, by the same measuring procedure, obtain different values for λLIV DSR (Deformed Relativistic Symmetries) The laws of motion are modified at a scale ℓDSR Different (inertial) observers, by the same measuring procedure, obtain the same value for ℓDSR (ℓDSR is a relativistic invariant as c) E2 − p2+ℓDSREp2 = m2 holds for all (inertial) observers
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV Coordinates of different (inertial) observers are connected by ordinary Poincar´ e transformations E2 − p2+λLIV Ep2 = m2 holds only for a preferred observer → Poincar´ e symmetries are broken Different (inertial) observers, by the same measuring procedure, obtain different values for λLIV DSR (Deformed Relativistic Symmetries) The laws of motion are modified at a scale ℓDSR Different (inertial) observers, by the same measuring procedure, obtain the same value for ℓDSR (ℓDSR is a relativistic invariant as c) E2 − p2+ℓDSREp2 = m2 holds for all (inertial) observers Coordinates of different (inertial) observers are connected by (ℓ-)deformed Poincar´ e transformations (nonlinear) → Poincar´ e symmetries are deformed
∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
Planck-scale modified dispersion relation E2 − p2+LpEp2 = m2
(Lp ∼ 10−19GeV)
LIV (Lorentz Invariance Violation) The laws of motion are modified at a scale λLIV Coordinates of different (inertial) observers are connected by ordinary Poincar´ e transformations E2 − p2+λLIV Ep2 = m2 holds only for a preferred observer → Poincar´ e symmetries are broken Different (inertial) observers, by the same measuring procedure, obtain different values for λLIV DSR (Deformed Relativistic Symmetries) The laws of motion are modified at a scale ℓDSR Different (inertial) observers, by the same measuring procedure, obtain the same value for ℓDSR (ℓDSR is a relativistic invariant as c) E2 − p2+ℓDSREp2 = m2 holds for all (inertial) observers Coordinates of different (inertial) observers are connected by (ℓ-)deformed Poincar´ e transformations (nonlinear) → Poincar´ e symmetries are deformed
It is well known that in the DSR approach the observable effects coming from modification of Poincar´ e symmetries are in general smoother, and some of the processes allowed by LIV are not allowed in DSR ∼ 2000-2001 Amelino-Camelia, Kowalski-Glikman,Magueijo,Smolin
The possibility of testing Planck-scale modified dispersion relation has been a central topic for quantum gravity phenomenology in the last two decades.
The possibility of testing Planck-scale modified dispersion relation has been a central topic for quantum gravity phenomenology in the last two decades. Especially through the observation of highly energetic astrophysical sources, for which the cosmological distance traveled by particles act as amplifier for the small, Planckian, effect.
The possibility of testing Planck-scale modified dispersion relation has been a central topic for quantum gravity phenomenology in the last two decades. Especially through the observation of highly energetic astrophysical sources, for which the cosmological distance traveled by particles act as amplifier for the small, Planckian, effect. You cannot neglect the effects of spacetime curvature/expansion →
FRW spacetime
For the LIV scenario, a consensus has been reached for the expected delay in the arrival of hard (high energy) photons respect to simultaneously emitted soft photons: ∆t = λLIV ∆p z 1 + ¯ z H (¯ z) d¯ z
∼ 2006 -2008
For the LIV scenario, a consensus has been reached for the expected delay in the arrival of hard (high energy) photons respect to simultaneously emitted soft photons: ∆t = λLIV ∆p z 1 + ¯ z H (¯ z) d¯ z
∼ 2006 -2008 Granot’s talk
This has allowed to put upper bounds on the scale of Lorentz symmetry breaking through observation of highly energetic transient astrophysical events: GRBs, AGN,... Fermi, HESS,... short GRBs:
λLIV ≤ 0.82Lp! GRB090510
For the LIV scenario, a consensus has been reached for the expected delay in the arrival of hard (high energy) photons respect to simultaneously emitted soft photons: ∆t = λLIV ∆p z 1 + ¯ z H (¯ z) d¯ z
DSR was only formulated in flat spacetime This was due mainly to technical and conceptual difficulties Interplay between redshift and deformation scale The recent developments of DSR research allowed us to fill this gap.
Ellis, Jacob, Piran,... ∼ 2006 -2008
1
Covariant Hamiltonian mechanics in de Sitter spacetime
2
DSR-deSitter
3
DSR-FRW: a glimpse
1
Covariant Hamiltonian mechanics in de Sitter spacetime
2
DSR-deSitter
3
DSR-FRW: a glimpse
We want to study the motion of particles in de Sitter spacetime: i.e. homogeneous isotropic universe expanding with a constant rate H
We want to study the motion of particles in de Sitter spacetime: i.e. homogeneous isotropic universe expanding with a constant rate H We rely on the (1+1D)
de Sitter algebra of symmetry generators {E, p} = Hp, {N, E} = p+HN, {N, p} = E C = E2 − p2 − 2HN
We want to study the motion of particles in de Sitter spacetime: i.e. homogeneous isotropic universe expanding with a constant rate H We rely on the (1+1D)
de Sitter algebra of symmetry generators {E, p} = Hp, {N, E} = p+HN, {N, p} = E C = E2 − p2 − 2HN Conformal time (η = H−1 1 − e−Ht ) canonical coordinates
{Ω, η} = 1, {Ω, x} = 0, {Ω, Π} = 0 {Π, η} = 0, {Π, x} = −1, {η, x} = 0
E = Ω(1 − Hη) + HxΠ, p = Π, N = xΩ(1 − Hη) − Π
2 η2 − H 2 x2
We want to study the motion of particles in de Sitter spacetime: i.e. homogeneous isotropic universe expanding with a constant rate H We rely on the (1+1D)
de Sitter algebra of symmetry generators {E, p} = Hp, {N, E} = p+HN, {N, p} = E C = E2 − p2 − 2HN Conformal time (η = H−1 1 − e−Ht ) canonical coordinates
{Ω, η} = 1, {Ω, x} = 0, {Ω, Π} = 0 {Π, η} = 0, {Π, x} = −1, {η, x} = 0
E = Ω(1 − Hη) + HxΠ, p = Π, N = xΩ(1 − Hη) − Π
2 η2 − H 2 x2
H = C − m2 = (1 − Hη)2 Ω2 − Π2 − m2 → 0 generates equations of motion ∂f/∂τ = {H, f}
We want to study the motion of particles in de Sitter spacetime: i.e. homogeneous isotropic universe expanding with a constant rate H We rely on the (1+1D)
de Sitter algebra of symmetry generators {E, p} = Hp, {N, E} = p+HN, {N, p} = E C = E2 − p2 − 2HN Conformal time (η = H−1 1 − e−Ht ) canonical coordinates
{Ω, η} = 1, {Ω, x} = 0, {Ω, Π} = 0 {Π, η} = 0, {Π, x} = −1, {η, x} = 0
E = Ω(1 − Hη) + HxΠ, p = Π, N = xΩ(1 − Hη) − Π
2 η2 − H 2 x2
H = C − m2 = (1 − Hη)2 Ω2 − Π2 − m2 → 0 generates equations of motion ∂f/∂τ = {H, f} Worldlines v(η)= dx(η)
dη
= dx/dτ
dη/dτ
{H,η}
x Η
v(η) =
Π
m2 (1−Hη)2
m→0
− → 1
x t
v(t) = e
−Htv(η(t)) m→0
− → e−Ht
Finite de Sitter translations (η, x)B = e−ξp ⊲ e−ζE ⊲ (η, x)A
eaG ⊲ x ≡ ∞
n=0 an n! {G, x}n,
{G, x}n = {G, {G, x}n−1}, {G, x}0 = x
Finite de Sitter translations (η, x)B = e−ξp ⊲ e−ζE ⊲ (η, x)A
eaG ⊲ x ≡ ∞
n=0 an n! {G, x}n,
{G, x}n = {G, {G, x}n−1}, {G, x}0 = x
ηB =eHζηA − eHζ −1 H , xB = eHζ(xA − ξ), ΩB = e−HζΩA, ΠB = e−HζΠA,
Finite de Sitter translations (η, x)B = e−ξp ⊲ e−ζE ⊲ (η, x)A
eaG ⊲ x ≡ ∞
n=0 an n! {G, x}n,
{G, x}n = {G, {G, x}n−1}, {G, x}0 = x
ηB =eHζηA − eHζ −1 H , xB = eHζ(xA − ξ), ΩB = e−HζΩA, ΠB = e−HζΠA, In the case a photon is emitted locally to Alice, the condition for Bob, distant T from Alice, to be along the photon worldline, is ζ = T and ξ =H−1(1−e−HT )
Finite de Sitter translations (η, x)B = e−ξp ⊲ e−ζE ⊲ (η, x)A
eaG ⊲ x ≡ ∞
n=0 an n! {G, x}n,
{G, x}n = {G, {G, x}n−1}, {G, x}0 = x
ηB =eHζηA − eHζ −1 H , xB = eHζ(xA − ξ), ΩB = e−HζΩA, ΠB = e−HζΠA, In the case a photon is emitted locally to Alice, the condition for Bob, distant T from Alice, to be along the photon worldline, is ζ = T and ξ =H−1(1−e−HT )
tA xA Bob tB xB Alice
1
Covariant Hamiltonian mechanics in de Sitter spacetime
2
DSR-deSitter
3
DSR-FRW: a glimpse
Consider the deformation of de Sitter Casimir C = E2−p2−2HNp + ℓ
Consider the deformation of de Sitter Casimir C = E2−p2−2HNp + ℓ
The ℓ-deformed (2D) de Sitter algebra of charges compatible with the invariance of C is {E, p} = Hp − ℓ(α − γ)HEp {N, E} = p + HN − ℓ(α − γ)E(p + HN) − ℓβEp {N, p} = E + 1 2 ℓ(α + 2γ)E2 + 1 2 ℓβp2
Consider the deformation of de Sitter Casimir C = E2−p2−2HNp + ℓ
The ℓ-deformed (2D) de Sitter algebra of charges compatible with the invariance of C is {E, p} = Hp − ℓ(α − γ)HEp {N, E} = p + HN − ℓ(α − γ)E(p + HN) − ℓβEp {N, p} = E + 1 2 ℓ(α + 2γ)E2 + 1 2 ℓβp2 We choose the following representation in terms of canonical conformal-time coordinates E =(1−Hη)Ω+HxΠ− ℓ 2 (α−γ)((1−Hη)Ω+HxΠ), p = Π, N = xΩ(1 − Hη) − Π
2 η2 − H 2 x2
2 β
ΩΠ + xΠ2 +ℓ 2 γx
Consider the deformation of de Sitter Casimir C = E2−p2−2HNp + ℓ
The ℓ-deformed (2D) de Sitter algebra of charges compatible with the invariance of C is {E, p} = Hp − ℓ(α − γ)HEp {N, E} = p + HN − ℓ(α − γ)E(p + HN) − ℓβEp {N, p} = E + 1 2 ℓ(α + 2γ)E2 + 1 2 ℓβp2 We choose the following representation in terms of canonical conformal-time coordinates E =(1−Hη)Ω+HxΠ− ℓ 2 (α−γ)((1−Hη)Ω+HxΠ), p = Π, N = xΩ(1 − Hη) − Π
2 η2 − H 2 x2
2 β
ΩΠ + xΠ2 +ℓ 2 γx
C =(1 − Hη)2 Ω2−Π2 +ℓ (1−Hη)3 γΩ3+βΩΠ2
Consider the deformation of de Sitter Casimir C = E2−p2−2HNp + ℓ
The ℓ-deformed (2D) de Sitter algebra of charges compatible with the invariance of C is {E, p} = Hp − ℓ(α − γ)HEp {N, E} = p + HN − ℓ(α − γ)E(p + HN) − ℓβEp {N, p} = E + 1 2 ℓ(α + 2γ)E2 + 1 2 ℓβp2 We choose the following representation in terms of canonical conformal-time coordinates E =(1−Hη)Ω+HxΠ− ℓ 2 (α−γ)((1−Hη)Ω+HxΠ), p = Π, N = xΩ(1 − Hη) − Π
2 η2 − H 2 x2
2 β
ΩΠ + xΠ2 +ℓ 2 γx
C =(1 − Hη)2 Ω2−Π2 +ℓ (1−Hη)3 γΩ3+βΩΠ2 In these coordinates photons move with velocity v(η) = 1−ℓ (γ + β) (1 − Hη) Π
We want to find the time of arrival of a hard photon respect to a soft photon emitted simultaneously at a distant source
We want to find the time of arrival of a hard photon respect to a soft photon emitted simultaneously at a distant source Consider two observers at relative rest. Alice local to the emission event, Bob local to the detector
We want to find the time of arrival of a hard photon respect to a soft photon emitted simultaneously at a distant source Consider two observers at relative rest. Alice local to the emission event, Bob local to the detector The worldlines described by Bob are xB(ηB) = xB
OA +
ηB
ηB
OA
dη vB(η) = xB
OA +(ηB−ηB OA) − ℓ(ηB−ηB OA)(γ+β)ΠB
1−H(ηB+ηB
OA)
Bob and Alice’s coordinates are connected by a finite translation, but now the action of the translation generators is deformed {E, η} = (1 − Hη) (1−ℓ(α − γ)E) , {p, η} = 0, {E, x} = −Hx (1−ℓ(α − γ)E) , {p, x} = −1.
Bob and Alice’s coordinates are connected by a finite translation, but now the action of the translation generators is deformed {E, η} = (1 − Hη) (1−ℓ(α − γ)E) , {p, η} = 0, {E, x} = −Hx (1−ℓ(α − γ)E) , {p, x} = −1. Exponentiating the Poisson brackets we find ηB = 1−eHζ H + eHζηA+ℓ (α−γ) ζeHζ(1−HηA)EA
H,ξ
xB = eHζ(xA − ξ)−ℓ(α−γ)ζeHζH(xA − ξ)EA
H,ξ
ΩB = e−Hζ ΩA+ℓ(α−γ)HζΩAEA
H,ξ
ΠA+ℓ(α−γ)HζΠAEA
H,ξ
H,ξ = ΩA
1 − HηA + HΠA xA − ξ
Bob and Alice’s coordinates are connected by a finite translation, but now the action of the translation generators is deformed {E, η} = (1 − Hη) (1−ℓ(α − γ)E) , {p, η} = 0, {E, x} = −Hx (1−ℓ(α − γ)E) , {p, x} = −1. Exponentiating the Poisson brackets we find ηB = 1−eHζ H + eHζηA+ℓ (α−γ) ζeHζ(1−HηA)EA
H,ξ
xB = eHζ(xA − ξ)−ℓ(α−γ)ζeHζH(xA − ξ)EA
H,ξ
ΩB = e−Hζ ΩA+ℓ(α−γ)HζΩAEA
H,ξ
ΠA+ℓ(α−γ)HζΠAEA
H,ξ
H,ξ = ΩA
1 − HηA + HΠA xA − ξ
and ξ =H−1(1−e−HT ) so that Bob, distant the comoving time T from Alice, detects the soft photon at time 0
Bob and Alice’s coordinates are connected by a finite translation, but now the action of the translation generators is deformed {E, η} = (1 − Hη) (1−ℓ(α − γ)E) , {p, η} = 0, {E, x} = −Hx (1−ℓ(α − γ)E) , {p, x} = −1. Exponentiating the Poisson brackets we find ηB = 1−eHζ H + eHζηA+ℓ (α−γ) ζeHζ(1−HηA)EA
H,ξ
xB = eHζ(xA − ξ)−ℓ(α−γ)ζeHζH(xA − ξ)EA
H,ξ
ΩB = e−Hζ ΩA+ℓ(α−γ)HζΩAEA
H,ξ
ΠA+ℓ(α−γ)HζΠAEA
H,ξ
H,ξ = ΩA
1 − HηA + HΠA xA − ξ
and ξ =H−1(1−e−HT ) so that Bob, distant the comoving time T from Alice, detects the soft photon at time 0 xB
OA = − eHT −1
H +ℓ(α−γ)T
ηB
OA = − eHT −1
H +ℓ (α−γ) TeHT ΠB
xB
OA = − eHT −1
H +ℓ(α−γ)T
ηB
OA = − eHT −1
H +ℓ (α−γ) T eHT ΠB
manifestation of
RELATIVE LOCALITY for translations: Alice, local to the event of emission, describes the photons to be emitted simultaneously in her origin. Bob, distant to the event of emission, describes the photons to be emitted in different points, depending
G.Amelino-Camelia+M.Matassa+F.Mercati+G.R.,Phys.Rev.Lett.106(2011)071301 G.Amelino-Camelia+N.Loret+G.R.,Phys.Lett.B700(2011)150 xA ηA detector emission xB ηB detector emission (ηB
OA, xB OA)
Problem of locality raised e.g. by Schutzhold+Unruh2003, Hossenfelder2010 Another discussion by Smolin2010
xB
OA = − eHT −1
H +ℓ(α−γ)T
ηB
OA = − eHT −1
H +ℓ (α−γ) T eHT ΠB
manifestation of
RELATIVE LOCALITY for translations: Alice, local to the event of emission, describes the photons to be emitted simultaneously in her origin. Bob, distant to the event of emission, describes the photons to be emitted in different points, depending
G.Amelino-Camelia+M.Matassa+F.Mercati+G.R.,Phys.Rev.Lett.106(2011)071301 G.Amelino-Camelia+N.Loret+G.R.,Phys.Lett.B700(2011)150
A general result for DSR theories: locality remains objective to observers local to the coincidence of events, but observers who are distant in their coordinatizion of spacetime “infer” those same pairs of events as not coincident Thus in DSR theories, by introducing an invariant inverse-momentum scale ℓ, one has to renounce to the concept of absolute locality in favour of a relative locality, just as one is enforced to abandon the idealization of absolut simultaneity by introducing a constant velocity of massless particles in a relativistic theory Relative locality affects only the coordinatization of distant events, not the observables. (measurements are local)
Summarizing Bob describes the trajectory xB(ηB) = xB
OA +(ηB−ηB OA) − ℓ(ηB−ηB OA)(γ+β)ΠB
1−H(ηB+ηB
OA)
xB
OA = − eHT −1
H +ℓ(α−γ)T
ηB
OA = − eHT −1
H +ℓ (α−γ) TeHT ΠB
Summarizing Bob describes the trajectory xB(ηB) = xB
OA +(ηB−ηB OA) − ℓ(ηB−ηB OA)(γ+β)ΠB
1−H(ηB+ηB
OA)
xB
OA = − eHT −1
H +ℓ(α−γ)T
ηB
OA = − eHT −1
H +ℓ (α−γ) TeHT ΠB We get the delay ∆ηB=ηB
h (xB h =0)=ℓpB h
2H
1
Covariant Hamiltonian mechanics in de Sitter spacetime
2
DSR-deSitter
3
DSR-FRW: a glimpse
How to generalize to FRW? ds2 = dt2 − a2(t)dx2 = ds2 = a2(η)
dt = a(η)dη
How to generalize to FRW? ds2 = dt2 − a2(t)dx2 = ds2 = a2(η)
dt = a(η)dη The expansion rate now depends on time H(t) =
1 a(t) da(t) dt
=
1 a2(η) da(η) dη
How to generalize to FRW? ds2 = dt2 − a2(t)dx2 = ds2 = a2(η)
dt = a(η)dη The expansion rate now depends on time H(t) =
1 a(t) da(t) dt
=
1 a2(η) da(η) dη
FRW is not a maximally symmetric spacetime, and in particular time translation symmetry is lost
How to generalize to FRW? ds2 = dt2 − a2(t)dx2 = ds2 = a2(η)
dt = a(η)dη The expansion rate now depends on time H(t) =
1 a(t) da(t) dt
=
1 a2(η) da(η) dη
FRW is not a maximally symmetric spacetime, and in particular time translation symmetry is lost Moreover we have to consider the relative locality effects: translations depend on momenta, and you cannot rely on a single observer’s description of a distant event.
How to generalize to FRW? ds2 = dt2 − a2(t)dx2 = ds2 = a2(η)
dt = a(η)dη The expansion rate now depends on time H(t) =
1 a(t) da(t) dt
=
1 a2(η) da(η) dη
FRW is not a maximally symmetric spacetime, and in particular time translation symmetry is lost Moreover we have to consider the relative locality effects: translations depend on momenta, and you cannot rely on a single observer’s description of a distant event. We cannot use directly translational symmetries to connect Bob’s with Alice’s coordinates
G.Amelino-Camelia+G.R., forthcoming
Slicing!
detector xBN tBN emitter Alice HN HN−1 HN−2 H1 H2 ... ...
We divide the time interval between the event of emission and the event of detection in
DSR-deSitter slices : in each slice particles move in a DSR-deSitter background, characterized by a different expansion rate Hn = H(tn) n = 1, . . . , N. We define a set of intermediate observers between Alice and Bob, one for each slice, so that the slices are carefully matched together. We recover the DSR-FRW trajectory in the limit N → ∞.
G.Amelino-Camelia+G.R., forthcoming
We finally get the delay, in terms of the redshift ∆t=ℓph z d¯ z H (¯ z) (β + γ) (1 + ¯ z) + α − γ 1+¯ z
z−H(¯ z) ¯
z(t)
d¯ z′ H (¯ z′)
G.Amelino-Camelia+G.R., forthcoming
We finally get the delay, in terms of the redshift ∆t=ℓph z d¯ z H (¯ z) (β + γ) (1 + ¯ z) + α − γ 1+¯ z
z−H(¯ z) ¯
z(t)
d¯ z′ H (¯ z′)
Compare with the LIV formula ∆t = λLIV ∆p z 1 + ¯ z H (¯ z) d¯ z λLIV ≤ 0.82Lp
G.Amelino-Camelia+G.R., forthcoming
We finally get the delay, in terms of the redshift ∆t=ℓph z d¯ z H (¯ z) (β + γ) (1 + ¯ z) + α − γ 1+¯ z
z−H(¯ z) ¯
z(t)
d¯ z′ H (¯ z′)
Compare with the LIV formula ∆t = λLIV ∆p z 1 + ¯ z H (¯ z) d¯ z
1 2 3 4 5 z 0.00 0.05 0.10 0.15 t p sec
Α Γ 1 , Β Γ 0 Α Γ 0 , Β Γ 1
The present bounds from GRBs do not apply to DSR! New analyses can be used to constrain DSR parameter space! → new methods? Granot λLIV ≤ 0.82Lp