My life at Caltech has been enriched by the presence of many wonderful friends. The graph on the right is an increase in the small range of α for which the first constraint in Eq.
Spontaneous Violation of Lorentz Invariance
- Chapter 2: Aether compactification
- Chapter 3: Instabilities in the aether
- Chapter 4: Sigma-model aether
- Chapter 5: Lorentz violation in Goldstone gravity
In the Friedmann-Robertson-Walker background dominated by the cosmological fluid, we find that the aether evolves dynamically to be quite time-like in the rest frame of the background fluid. Kraus and Tomboulis examined this possibility in the case of the photon and found that it was feasible.
Inflation
Chapter 6: Unitary evolution and cosmological fine-tuning
The criterion is derived from the Hamiltonian (symplectic) structure of general relativity, and has the nice properties that it is (i) independent of the choice of time slice, (ii) is always positive, and (iii) the underlying symmetry of the theory . The extra dimensions can be "large" if the expected value of the ether field is much larger than the inverse coupling.
Aether
However, the mass distributions, which are different for different species, are general predictions of the model. In a sense, the effect of the aether field is a distortion of the background metric, but in a way that is felt differently by different types of fields.
Energy-Momentum and Compactification
The non-vanishing expectation value of the aether field does not itself produce any energy density. In the context of an extra dimension, this implies that the ether field will not make a contribution to the effective potential of the radio, which is why the task of stabilizing the extra dimension must be left to other mechanisms.
Scalars
With the addition of the aether field, the mass spacing between the different states in the KK tower is improved. The parameter αφ is a ratio of the ether vev to the mass rate μφ that characterizes the coupling and can be much larger than unity.
Gauge Fields
Fermions
Although the form of this equation is the same as the scalar case and the gauge field case, it is quantitatively different: for large α, the improvement goes as α4 and not as α2. Similar to the scalar case, if we do not impose Z2symmetry, we will have to consider the following two lower-order couplings: uaψγ¯ aψ and µiuaψ∂¯ aψ.
Gravity
In this decomposition, ¯hµν represents the propagating gravitational waves, Ψ the Newtonian gravitational fields, and Φ the radion field representing the breathing mode of the extra dimension. We have already argued that there will be no macroscopic deviations from Newton's law on the scale of the extradimensional radius R because the zero-mode fields are uniformly distributed throughout the extra dimensions.
Conclusions
In the Maxwell case, the Hamiltonian is unbounded below; however, a perturbative analysis reveals no explicit instabilities in the form of tagyons or ghosts. Lim [18] calculated the Hamiltonian for small perturbations around a constant time-like vector field in the rest frame, and derived constraints on the coefficients of the kinetic terms.
Models
Validity of effective field theory
In the presence of Lorentz violation, therefore, the field of validity of the effective field theory can be significantly reduced in highly enhanced frameworks. We have been careful to include all the lowest-order terms in the effective field theory expansion—the terms in (3.8).
Boundedness of the Hamiltonian
- Timelike vector field
- Spacelike vector field
- Smooth potential
- Discussion
Expressed in terms of the variables φ, θ, ψ, the Hamiltonian is a function of the initial data that automatically obeys the fixed rate constraint. We have seen that the Hamiltonian is unbounded from below unless the kinetic term takes the form of the sigma pattern, (∂µAν) (∂µAν).
Linear Instabilities
Timelike vector field
We have shown that when the phase velocity of a given field excitation is greater than the speed of light in a preferred rest frame, there exists a (highly amplified) frame in which the excitation appears unstable - that is, the frequency of the field excitation. can be imaginary. In appendix A we find dispersion relations of the form in (3.54) for the various massless excitations around a constant time-like background (tµ= ¯Aµ/m).
Spacelike vector field
Stability is not frame-dependent
Consider amplifying the wave-four vectors of such excitations with complex-valued frequencies and real-valued spatial wavevectors back to the rest frame. Then, in the rest frame, both the frequency and the spatial wave vector will have non-zero imaginary parts.
Negative Energy Modes
Spin-1 energies
In this section we consider non-vanishing β4 and show that the spin-1 mode can transport negative energy even when the conditions for linear stability are met. In both the time-like and space-like cases, models with β4 6= 0 have spin-1 modes that can be ghostly.
Spin-0 energies
We have therefore shown that, for time-like backgrounds, there are modes which in some frame have negative energies and/or growing amplitudes as long as β1 6=β∗, β1 6= 0 and β∗ 6= 0. Again, the Hamiltonian density is less than zero for modes with sufficiently long wavelengths (k < e−3|η|m), so the effective theory is valid.
Maxwell and Scalar Theories
Maxwell action
It is possible that this obstacle to a well-defined evolution will be governed by higher-order terms in the effective field theory. For the time-like case, Seifert found a gravitational instability in the presence of a spherically symmetric source [50].
Scalar action
To satisfy the fixed rate constraint, the spatial components of the vector field would have to be imaginary, which is unacceptable since Aµ is a real-valued vector field. If the higher-order terms in the effective action have time derivatives of the Ai components, these terms may become important for the dynamical evolution of the vector field, indicating that we have left the realm of validity of the low-energy effective field theory that we are considering.
Conclusions
The spin-2 mode can propagate subluminally for some values of the vector field/Ricci tensor coupling;. Finally, we consider the cosmological evolution of the vector field in two different backgrounds.
Excitations in the Presence of Gravity
We study the evolution of a time-like vector field in a generalized flat Friedmann-Robertson-Walker (FRW) background and find that the vector field tends to align to be perpendicular to hypersurfaces of constant density. The pole is physical if v > 1 and (in the limit as ˆn·ˆk→ 0) when β passes through the pole (β2 →β2 > v−2), the frequency acquires a non-zero imaginary part corresponding to the increasing amplitude mode. The frequency becomes imaginary at some β2 <1 as long as ˆn·kˆ6= 1.) The time scale on which it grows is set by 1/Im(ω).
Experimental Constraints
The strongest constraint in the α < −1 region is from Eq. 4.18), and for most of the α > −1 region, the strongest constraint from the second inequality in Eq. The plot on the right is a blow-up of the small range of α for which the first constraint in Eq.
Cosmological Evolution
This is a reasonable assumption given that the background space is homogeneous and therefore should only affect the time evolution of the vector field. We can use the rotational invariance of the FRW background to choose coordinates such that the x-axis is aligned with the spatial part of the vector field.
Extra Dimensions
We conclude that a time-like vector field will generically tend to be completely time-like in the rest of the cosmological fluid, thus restoring the isotropicity of the cosmological background. However, when the universe enters an expansion phase where a(t) =t2/3(1+w) and w is strictly greater than −1 (and less than 1), then the component of the vector field in the fifth dimension will collapse.
Conclusions
Then two linear combinations of the Goldstone states have exactly the same properties as the photon in electromagnetism. This is reminiscent of the photon case, where a longitudinal mode (in addition to the two transverse modes) becomes dynamical in the presence of the radiation corrections induced by integrating the massive modes.
Goldstone Electromagnetism
- Photons as Goldstone bosons
- Radiative corrections and dispersion relations of the Goldstone modes 88
- Gravitons as Goldstone bosons
- Radiative corrections and dispersion relations
Expressing the longitudinal mode in the kµ and ¯Aµ basis automatically makes it orthogonal to the transverse modes. However, in the presence of radiative correction terms, we expect them to become dynamic, similar to the longitudinal mode in the vector case.
Anisotropic Propagation
- Dispersion relations
- Motion of test particles
- Experimental constraints
- Corrections to the energy-momentum tensor
The correction of the dispersion relation also has an effect on the energy momentum tensor of the transverse Goldstone modes. With the change of the dispersion relation of the gravitons, k µ changes as k µ → k µ+c22H µνkν up to the first order.
Vevs That Do Not Break All Six Generators
Gravitons are not necessarily Goldstone
A similar story holds in the case of graviton, as long as all six generators of the Lorentz group are broken, creating six Goldstone bosons. In this section we will explore what happens if all six generators are not destroyed by the vev.
An example: Three Goldstone bosons only
The equations of motion (5.57) have three zero eigenvalues, which is consistent with the fact that there are three degrees of freedom of the residual gauge. There are also only two mass modes, since vev creates only two independent cardinal gauge conditions.
Conclusions
The purpose of inflation is to make the conditions of the hot, dense, quiet Big Bang seem natural. The homogeneity of the universe represents a literal true tuning; there is no reason for the universe to be still.
The Evolution of Our Comoving Patch
Autonomy
Time-slicing allows us to view the universe as a fixed set of degrees of freedom that evolve over time, obeying Hamilton's equations. Our moving spot is defined by the interior of the intersection of our previous light cone with a cut-off surface, for example the surface of the last scattering.
Unitarity
We cannot definitively answer this question in the absence of a theory of quantum gravity, but for the purposes of this chapter we will assume that the space of states is not expanding. This distinction between the number of states implied by the unitarity assumption and the number of states that can reasonably be described by quantum fields on a solid background is absolutely crucial to the question of how precise are the conditions necessary to trigger inflation.
The Canonical Measure
Gibbons, Hawking and Stewart (GHS; [73]) showed how the Liouville measure on the phase space could be used to define a unique measure on the solution space (see also. It is immediately clear that the measure on this set is independent of the choice of the Hamiltonian.
Flatness
The measure is defined by choosing some transverse surface Σ, and integrating the dotted B-field in an orthogonal one-formni. We feel that this interpretation is the most sensible one, even though it conflicts with the conventional presentation of the flatness problem.2 The usual statement of the flatness problem notes that even a very small departure from flatness in early times leads to a significant amount of growth. of curvature at late times.
Homogeneity
Action for a perfect fluid background
The dynamical quantity for hydrodynamic matter is ξα(xβ), the deviation of the test particles from their trajectory in the unperturbed FRW universe. The fact that we can express the action in terms of the Mukhanov-Sasaki variable v alone implies that only one dynamical degree of freedom is present.
Action during inflation
Computation of the measure
The extended Hamiltonian, H+=p2v/2−(c2sk2−z00(t)/z(t))v2− H, is explicitly time-independent (identically zero), and its conservation is analogous to the constraint of Eq. of Friedman in the analysis of the flattening problem. The flux of trajectories traversing this surface is unity, the coefficient of the first term in (6.51).
Tracing Perturbations Backwards
- Relation to Planck-scale cutoffs
- Evolution of perturbations in a matter-dominated universe
- Evolution of perturbations in a matter-dominated universe preceded
- Results
The perturbation-invariant gauge energy densityδρfi can now be calculated by substituting Ψ and Ψ0 into (6.70). With inflation, in contrast, we can start with the Hubble parameter at the Planck scale and any sub-Planckian value of the perturbations.
What is Inflation Good For?
The universe is not chosen randomly
We believe that the benefit of inflation is not that it makes universes like ours the most numerous in the space of all possible universes, but that it provides a more reasonable target for a true theory of initial conditions, from quantum cosmology or elsewhere . In contrast, most initial states growing in our universe today show no signs of being ready to do so at early times; there is no way of knowing what they are.
Inflation as an easy target
Then, in less detail, we find the solutions of the equations of motion linearized with respect to the space-like background. Three independent solutions of these equations are given by setting the eigenvalue of the matrix M to zero and setting c to the corresponding eigenvector.
Polarizations of Goldstone Modes
- Time-time
- Time-space
- Diagonal space-space
- Off-diagonal space-space
Finally, we consider the case in which one of the off-diagonal spatial components is nonzero. We can, for example, perform a rotation to diagonalize the three modes in B.1.4 so that they become a linear combination of the modes in B.1.3.
Proof That Gravitons Can Be Goldstone Bosons
The two transverse linear combinations of the six Goldstone modes . 170
We now proceed to show that two linear combinations of the Goldstone modes (M5 and M6) obey the dispersion relation kµkµ= 0 and are transverse to momentum (kµhµν = 0). In summary, we have shown that there are two special linear combinations (M5 and M6) of the six Goldstone states that have a polarization tensor identical to that of a graviton in general relativity; obey the normal dispersion relationk2 = 0; and is across the momentum kµ.
Proof of the Necessity of Breaking All Six Generators to Get Goldstone Gravi-
Since the rows of A are only linear combinations of those of N, the rank of the first is necessarily less than or equal to the second. This means the lack of two linear combinations of Goldstone modes that behave like a graviton in general relativity.
