Modified Dispersion Relations in Quantum Gravity

• Modified Dispersion Relations are deformations of the standard energy–momentum relation that incorporate Planck-suppressed higher-order terms, altering kinematics and reaction thresholds.
• They serve as a framework in quantum gravity phenomenology, Lorentz violation studies, and cosmological models to introduce measurable corrections like energy-dependent propagation speeds.
• Formulated via Hamilton geometry and effective field theories, MDRs enable model discrimination through analyses of astrophysical observations, gravitational waves, and high-energy cosmic rays.

Modified dispersion relations (MDRs) generalize the standard relativistic energy–momentum relation by including additional, often Planck-suppressed, higher-order and/or nonlinear terms in momentum or energy. MDRs arise ubiquitously in quantum gravity phenomenology, models of Lorentz invariance violation, studies of Finsler and Hamilton geometry, and effective descriptions of new-physics media. Their consequences are wide-ranging: they impact kinematical processes, introduce corrections to astrophysical and cosmological observables, and provide testable signatures of new physics such as quantum gravity or exotic matter sectors.

1. Formalism and Model Classes

Modified dispersion relations are commonly written as deformations of the standard relation $E^2 = p^2 + m^2$ (with $c = 1$):

$E^2 = p^2 + m^2 + \sum_{n>2} \eta_n\, p^n,$

or more generally, as

$E^2 f^2(E/E_P) - p^2 g^2(E/E_P) = m^2,$

where $E_P$ is the Planck scale, $\eta_n$ are dimensionful (often Planck-suppressed) coefficients, and $f,g$ are so-called rainbow functions capturing the energy dependence (Das et al., 2021).

Alternatively, for Lorentz-violating scenarios, species-dependent "refractive indices" may be introduced:

$$E^2 = m^2 n^{-4} + p^2 n^{-2}, \qquad n = 1 + \mathcal{A} E^\alpha,$$

with $\mathcal{A}$, $\alpha$ encoding the magnitude and scaling of the departure from Lorentz invariance (Artola et al., 24 Oct 2024).

In gravitational or cosmological backgrounds, MDRs are formulated as Hamilton functions $H(x,p)$ on the cotangent bundle $T^*M$, i.e.,

$H(x,p) = \text{modified function of } x, p,$

with the standard case corresponding to a quadratic form in $p$ dictated by the local metric, and MDRs corresponding to non-quadratic or non-homogeneous forms (Barcaroli et al., 2016, Barcaroli et al., 2015).

Specific model classes include:

Model/Famework	MDR Structure/Key Feature
Effective Field Theory (EFT)	$E^2 = p^2[1 + \kappa (p/M_P)^n] + m^2$; $n$ integer (0806.3496)
DSR/$\kappa$-Poincaré	Nonlinear energy terms: $\sim \sinh^2(\ell p_0/2)$, $e^{\ell p_0}p^2$ (Barcaroli et al., 2016, Barcaroli et al., 2015)
Finsler/Hamilton geometry	Hamiltonian $H(x,p)$ generally non-quadratic; intertwining geometry (Barcaroli et al., 2015)
Hořava–Lifshitz gravity	$E^2 \sim p^2 + \beta p^4 - \alpha p^6 + \ldots$ (Davies et al., 2023, Vacaru, 2010)
Extra-dimensional set-ups	$E^2 - p^2 \sim a L_P E^4 + d'L_P E^6 + \ldots$ (Sefiedgar et al., 2010)
Medium/dark medium effects	$E^2 = (1 + a)k^2 + 2bk$, $n(k) = 1 + M^2/k^2$ (for neutrino) (Masina et al., 2011)

2. Physical and Mathematical Implications

Kinematics and Thresholds:

MDRs modify the phase velocities for different particle species, which can open new reaction channels–notably, one-particle processes like vacuum Cherenkov radiation ($a \to a + \gamma$) and photon decay ($\gamma \to e^+ e^-$) (Balek et al., 2018, Artola et al., 24 Oct 2024).

Example:

A general MDR for species $a$ and $b$,

$$E_a^2 = m_a^2 n_a^{-4} + p_a^2 n_a^{-2}, \quad n_a = 1+\mathcal{A}_a E_a^\alpha,$$

permits processes forbidden in Lorentz-invariant kinematics when the phase velocity of $a$ exceeds that of $b$. The threshold and angular conditions for emission (e.g., for vacuum Cherenkov radiation) are determined by a small parameter $\Theta_c$, with emission allowed for $\Theta_c > 0$ (Artola et al., 24 Oct 2024).

Propagation effects:

MDRs generically induce frequency/energy-dependent propagation velocities ($v_{\rm ph}(k)$), leading to experimentally accessible consequences:

• Time-of-flight (lateshift) effects: The arrival time of photons (or other particles) from distant sources becomes energy-dependent, as encoded in general parametrized lag–redshift relationships (Caroff et al., 20 Dec 2024, Pfeifer, 2018).
• Redshift corrections: In FLRW or curved backgrounds, the observed cosmological redshift becomes momentum-dependent (Barcaroli et al., 2016, Pfeifer, 2018).

Quantum field and thermodynamic effects:

In systems sensitive to the field vacuum structure, MDRs modify:

• The density of states, as occurs for the photon thermal spectrum in the CMB, potentially explaining anomalies such as EDGES (Das et al., 2021).
• The response of accelerated detectors, with the spectrum modified still retaining Planckian structure for superluminal MDRs, but possibly exhibiting pathologies for subluminal cases (Davies et al., 2023).
• Corrections to black hole thermodynamics: the temperature and entropy can receive subleading corrections, including universal logarithmic terms, and may admit a limiting temperature that prevents full black hole evaporation (0807.4269).

Astrophysical PDEs and high-energy phenomena:

In the context of ultra high energy cosmic rays, observations constrain the energy loss due to vacuum Cherenkov processes, leading to extremely tight bounds on LIV/MDR parameters (Balek et al., 2018, Artola et al., 24 Oct 2024).

3. Hamilton Geometry and Phase Space Structures

MDRs are systematically formulated in Hamilton geometry, where the Hamiltonian $H(x,p)$ defines both particle dynamics and phase space geometry (Barcaroli et al., 2015):

• Metric-induced case: $H_g(x,p) = g^{ab}(x) p_a p_b$ yields flat momentum space and curved spacetime.
• Non-metric MDRs: Non-homogeneous $H(x,p)$ implies intertwined curvature in both spacetime and momentum space, i.e., the phase space curvature $R_{cab}(x,p)$ depends on both $x$ and $p$.
• For specific MDRs inspired by $q$–de Sitter or $\kappa$–Poincaré quantum groups, either the spacetime curvature becomes momentum dependent, or the momentum space itself is curved, directly encoding the Planck-scale modifications expected in quantum gravity.

4. Observational Signatures and Constraints

4.1. High-Energy Astrophysics and Cosmic Rays

• Vacuum Cherenkov and photon decay: Allowed regions in MDR parameter space are tightly constrained by non-observation in cosmic rays and high-energy photon events. For arbitrary $p^n$ corrections, the allowed region is a "wedge–band" region in $(\xi, \eta)$ parameter space, robust against changes in $n$, but strongly stretched (Balek et al., 2018, Artola et al., 24 Oct 2024).

4.2. Gravitational Wave Astronomy

• Phase dephasing: Lorentz-violating MDRs for gravitons accumulate a shift in GW inspiral signals. Constraints from LIGO/Virgo are tightest for MDR corrections with small powers of $p$ (low $\alpha$ in $E^2 = p^2 + \mathcal{A} p^\alpha$) and deteriorate at higher scaling (Mirshekari et al., 2011).

4.3. Black Hole Imaging, Shadows, and Lensing

• Photon sphere, black hole shadow, Shapiro delay, deflection angle: MDRs induce a dependence of these features on photon energy and angular momentum, which does not exist in general relativity. Observationally, this implies energy-dependent ("rainbow") features in black hole shadows and gravitational lensing events, accessible to high-resolution VLBI or lensing surveys (Läänemets et al., 2022, Pfeifer, 2019).

5. Cosmology and Early Universe

5.1. Inflation and Perturbations

• Scale-invariant spectra from MDRs: Inclusion of high-power (e.g., $k^6$) terms in the MDR can generate an exactly scale-invariant vacuum spectrum for cosmological perturbations, which may account for the observed CMB spectrum given suitable background dynamics—inflation with a sufficiently high Hubble rate is required to preserve this scale invariance at observable scales (Bianco et al., 2016).
• Trans-Planckian effects and power spectrum modifications: MDRs can induce "super-excited" or "calm excited" initial states, modifying or leaving unaffected the inflationary power spectrum depending on parameter choices. The presence of a negative-slope region in the dispersion leads to large corrections ("super-excitations"), while other regions can mimic the Bunch–Davies state ("calm excited states") (Ashoorioon et al., 2017).

5.2. Thermodynamics of FRW Universes

• GUP–MDR equivalence: In extra-dimensional cosmologies, MDRs and generalized uncertainty principles (GUP) yield equivalent corrections to horizon entropy, suggesting they are different manifestations of a minimal observable length in quantum gravity (Sefiedgar et al., 2010).

5.3. CMB and 21-cm Anomalies

• Density of states corrections: The density of photon states is modified, changing the CMB spectrum in the Rayleigh–Jeans regime. This can lead to larger-than-expected 21-cm absorption (e.g., the EDGES anomaly). The correction factor $R(E) = 1/[F^3(E/E_P)] |1 - (E F'(E/E_P)/F(E/E_P))|$ quantifies the deviation (Das et al., 2021).

6. Model Discrimination and Parameterization Strategies

A general Hamiltonian approach allows construction and discrimination of MDR models in cosmological symmetry. The parameterized lag–redshift relation

$$\Delta t = \frac{I+1}{2} \frac{(E_1^I - E_2^I)}{\Lambda^I} \int_0^z \frac{\sum_{m} B_m[t(z')] (1+z')^m}{H(z')} dz',$$

encompasses previous models (e.g. Jacob–Piran, $\kappa$–Poincaré, SpaM) and highlights the dependence of observables on both the MDR parameters and the source redshift coverage (Caroff et al., 20 Dec 2024). Discriminating among models requires a wide and flat redshift distribution of astrophysical sources (GRBs, AGN), as differences in the lag function $\kappa(z)$ only become evident when data covers a broad range.

7. Interpretational Ambiguities and Outlook

MDRs may signal quantum gravity effects, exotic matter backgrounds (e.g., dark medium), or propagating-medium-induced corrections (as in plasmas). The same formal apparatus describes either scenario, emphasizing the need for a careful comparison of experimental results across multiple probes, as well as a detailed theoretical investigation of possible degeneracies (Läänemets et al., 2022, Masina et al., 2011). High-precision multi-messenger astronomy, cosmological observations, and laboratory searches all contribute, with current constraints on MDR parameters being extremely tight—of order $|\mathcal{A}p^\alpha| < 10^{-17}$ for the highest energy cosmic rays (Artola et al., 24 Oct 2024).

In summary, the paper of modified dispersion relations provides a unifying language to encode quantum gravity phenomenology, Lorentz violation, and medium effects, offers a spectrum of novel observable predictions, and motivates an extensive multi-modal experimental program.