Gravitational Wave -- Gauge Field Dynamics
GGravitational Wave – Gauge Field Dynamics
R. R. Caldwell ∗ and C. Devulder † Department of Physics and Astronomy, Dartmouth College,6127 Wilder Laboratory, Hanover, NH 03755 USA
N. A. Maksimova ‡ Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA (Dated: June 5, 2017)
Abstract
The dynamics of a gravitational wave propagating through a cosmic gauge field are dramat-ically different than in vacuum. We show that a gravitational wave acquires an effective mass,is birefringent, and its normal modes are a linear combination of gravitational waves and gaugefield excitations, leading to the phenomenon of gravitational wave – gauge field oscillations. Thesesurprising results provide insight into gravitational phenomena and may suggest new approachesto a theory of quantum gravity.
Essay written for the Gravity Research Foundation 2017 Awards for Essays on Gravitation. ∗ [email protected] (corresponding author) † [email protected] ‡ [email protected] a r X i v : . [ g r- q c ] J un eneral relativity is ever full of surprises. Over one hundred years since its inception[1], the frontiers of gravitation remain fertile subjects. The first detection of gravitationalwaves has established the field of gravitational wave astronomy [2]. The search is on for theimprint of relic gravitational radiation of quantum origin on the polarization pattern of thecosmic microwave background [3], and the accelerating cosmic expansion hints at quantumgravitational effects [4, 5]. In this milieu, cosmic gauge fields have been widely investigatedfor a possible role in catalyzing an inflationary epoch [6–9]. As we present in this essay,these investigations have revealed novel properties of the gravitational wave - gauge fieldsystem [10].The key element in this work is a collection of non-Abelian gauge fields with a vacuumexpectation value (vev). The action for the theory is S = (cid:90) d x √− g (cid:18) M P R − F aµν F aµν (cid:19) , (1)where we use metric signature − + ++ and M P is the reduced Planck mass. In the simplestrealization, we consider an SU(2) gauge field with field strength tensor F aµν ≡ ∂ µ A aν − ∂ ν A aµ − g Y (cid:15) abc A bµ A cν , where g Y is the coupling constant. Greek letters represent space-timeindices, and Latin letters i, j, ... are spatial indices. The SU(2) indices are indicated by a, b, ... = 1 , ,
3, and are raised and lowered by a metric η = diag(1 , , A aµ with stress-energy that is isotropic andhomogeneous, consistent with the symmetries of the cosmological Robertson-Walker space-time. In a coordinate system ds = a ( τ ) ( − dτ + d(cid:126)x ), we adopt the Ansatz A bi = φ ( τ ) δ bi withall other components vanishing. In essence, we have identified the global part of the SU(2)with the O(3) rotational symmetry of spacetime. This “flavor-space locked” field configu-ration [11] resembles a pair of uniform electric and magnetic fields for each flavor, pointingalong the x − , y − , z − directions. Although the configuration is anisotropic in flavor, it isisotropic in pressure and energy. The equation of motion ∇ µ F aµν + g Y (cid:15) abc A µb F cµν = 0 reducesto φ (cid:48)(cid:48) +2 g Y φ = 0 under our Ansatz, which is solved in terms of elliptic Jacobi functions. Theclassical field amplitude simply oscillates, and the flavor-space locked configuration underthis model is stable, as shown through a linear perturbation analysis [12, 13].The gauge field strength tensor F aµν has non-zero components where we expect to findan electric field, “ E ” = F a i = φ (cid:48) δ ai /a . Due to the coupling g Y there is also a magnetic field,“ B ” = F aij = − g Y φ (cid:15) aij , which, for each flavor, is coaligned with the electric field. This2ev enables the gauge field to support transverse, traceless, synchronous tensor fluctuationswhich couple to gravitational waves.In order to build intuition, we first consider a monochromatic gravitational wave propa-gating along the z-direction. We choose a circularly polarized gravitational wave, becausethe gauge field has a built-in right handedness in the group structure constants (cid:15) ijk . As itsqueezes and stretches the gauge field along alternate axes in the x − y plane, it enhancesthe B field by an amount that is proportional to the gravitational wave amplitude, in phasewith the wave. The E field is also enhanced, but lags by π/
2. These arguments suggest the B and E act like a spring and an anti-spring.Second, we consider that the gauge field itself may fluctuate. By perturbing the fullgauge field equation of motion we note that fluctuations of the field A aµ in each directionenhance the E and B energy, preferentially in a right-handed circular pattern. This, ofcourse, reflects the built-in right-handedness. These rough arguments suggest that left- andright-circularly polarized fluctuations of the gauge field will propagate differently.Third, by perturbing the gravitational field and the gauge field simultaneously, we deducethat the wave-like excitations couple. That is, a monochromatic gravitational wave canproduce a wave-like excitation of the gauge field with the same wave number, and viceversa. This observation harks back to a remarkable series of papers starting with the workof Gertsenshteyn [14], in which the authors showed that a gravitational wave propagatingthrough a stationary magnetic field converts into an electromagnetic wave and back again[15–17]. In contrast to electromagnetism, the presence of three flavors allows us to build anisotropic medium.Here we investigate the more general phenomenon of the conversion of a gravitationalwave into a gauge field, as may be present in the early stages of the Universe. In particular,we discover that gravitational waves transform into gauge field waves, disappearing andreappearing much like neutrino flavor oscillations.We consider linearized gravitational waves and tensor fluctuations of the gauge field.Following a standard calculation, δg ij = a h P e Pij where P = ( L, R ) labels the circularpolarization and e Pij is the standard polarization matrix. Similarly, we consider δA aj = a y P e P aj which makes use of the same polarization matrix. A change of variables, h = H √ /aM P and y = Y / √ a , puts the action into canonical form. The equations of motion,3n terms of the Fourier amplitudes, are H (cid:48)(cid:48) L + (cid:20) k − a (cid:48)(cid:48) a + 2 a M P ( g Y φ − φ (cid:48) ) (cid:21) H L = 2 aM P (cid:2) ( g Y φ + k ) g Y φ Y L − φ (cid:48) Y (cid:48) L (cid:3) (2) Y (cid:48)(cid:48) L + (cid:2) k + 2 g Y kφ (cid:3) Y L = 2 aM P (cid:20) a (cid:18) φ (cid:48) a H L (cid:19) (cid:48) + g Y φ ( k − g Y φ ) H L (cid:21) . (3)The equations for H R , Y R are obtained by replacing k → − k . First, we observe, nowquantitatively, that the gravitational wave equation acquires a time-dependent, mass-liketerm m = 2 a M P ( g Y φ − φ (cid:48) ) (4)proportional to B − E , arising from the stress of the gauge field. Apart from the possi-bility that dominance of E or B can enhance or suppress a spectrum of long wavelengthgravitational waves, there is a deeper point to be made. The effective mass term introducesa new scale into the system, and the gravitational wave amplitude is no longer comparableto a massless, minimally coupled scalar field [18]. This feature may suggest new approachesto the issues facing theories of massive gravity: the background matters. That is, a cos-mological, spin-1 field may play an important role in a symmetry breaking scheme for thegraviton.Second, the coupled system is birefringent as revealed by the equations of motion forthe left- and right-circular polarizations. In particular, the dispersion term k + 2 g Y kφ canbe negative for a range of wavenumbers; this holds for one polarization but not the other,thereby preferentially amplifying the gauge field and consequently the gravitational waves.This chiral asymmetry could have profound implications for the search for the imprint ofprimordial gravitational waves on CMB polarization anisotropy. It means there should bea unique, parity-odd correlation between temperature and the so-called “B-modes” [19–21].This signal is already the target of current and planned experiments [22–29].Third, at high frequencies the gravitational wave and gauge field interconvert throughthe phenomenon of gravitational wave – gauge field oscillations [10]. This is more easilyseen if we focus on the Lagrangian for gravitational and gauge field waves propagating withFourier wavenumber k greater than the expansion rate or rate of change of the gauge field, (cid:76) = 12 H (cid:48) L − k H L + (cid:78) (cid:88) n =1 (cid:20) Y (cid:48) Ln − k Y Ln − kg Y φY Ln + 2 aM P H L (cid:0) kg Y φ Y Ln − φ (cid:48) Y (cid:48) Ln (cid:1)(cid:21) . (5)4 igure 1. The oscillations of the gravitational wave amplitude (cid:104) (black) and gauge field (cid:121) (dashed)are shown for (cid:78) = 1 , | ψ | = 1 and the oscillation period has been scaled. Thebottom right panel shows that the squared amplitudes sum to unity. (Figures reproduced fromRef. [10].) We have now allowed (cid:78) families of SU(2), embedded in a larger SU(N), where (cid:78) = [ N/ (cid:78) + 1 coupled oscillators. At highfrequency, H and each of Y n oscillate with frequency k . The gravitational wave couples toeach gauge field wave; each gauge field wave couples only to the gravitational wave.The gravitational wave – gauge field oscillations are revealed by the rms amplitude of thewaves in the high frequency limit. We write H = (cid:104) e − ikτ and Y n = (cid:121) n e − ikτ and choose k tobe sufficiently large such that we can treat the coefficients φ, φ (cid:48) as constants. The normalmodes are identified by diagonalizing the Lagrangian (5). Starting with the gravitationaland gauge field modes ψ i = ( (cid:104) , (cid:121) n ) for i = 0 , , ... , (cid:78) , we write the Lagrangian in theform (cid:76) = ψ (cid:48)† I ψ (cid:48) − ψ † M ψ . We transform into the eigenbasis of M via ψ i = R ij ∆ j ,where ∆ j = (∆ , ∆ n ) are the normal modes. Hence, the Lagrangian acquires the form (cid:76) = ∆ (cid:48)† I ∆ (cid:48) − ∆ † Ω ∆ and Ω is diagonal with the normal mode frequencies.Conservation of the canonical stress-energy tensor Θ µν = ∂ µ ψ i δ (cid:76) /δ∂ ν ψ i − η µν (cid:76) yields theconstant of motion in the high frequency limit, | ψ | = | (cid:104) | + (cid:80) (cid:78) n =1 | (cid:121) n | = (cid:80) (cid:78) n =0 | c n | , wherethe coefficients c are the initial values of (cid:104) , (cid:121) n . Hence, we determine that the gravitational5 igure 2. The gravitational wave – gauge field oscillations for the case (cid:78) = 3 are illustratedusing oscillating pistons. The gravitational and gauge field waves are represented by the centraland surrounding pistons. (Figure reproduced from Ref. [10]. View in Adobe Reader to playanimation.) and gauge field wave amplitudes trace a pattern on the surface of an (cid:78) + 1-dimensionalsphere, endowing the system with an emergent symmetry. This behavior is reminiscent ofneutrino flavor oscillations, where the mass eigenstates remain in phase while the flavoreigenstates oscillate. Examples of the oscillation patterns are shown in Fig. 1 and as ananimation in Fig. 2.The gravitational wave – gauge field oscillations have implications for the quantizationof the gravitational field. For example, in a gauge-field inflationary scenario, the Hilbertspace must be expanded to include the gauge field excitations. Quantum fluctuations inthe gravitational field give rise to a homogeneous solution H h , whereas fluctuations in thegauge fields create n = 1 , ..., (cid:78) inhomogeneous solutions H i, n . The power spectrum is (cid:104) H (cid:105) = | H h | + (cid:80) (cid:78) n =1 | H i, n | , which reflects the new emergent symmetry. The modulation ofthe individual amplitudes now cancels, yielding a constant amplitude | ψ | which is reflectedin the observable power spectrum.The example investigated in this essay has afforded us a deeper understanding of gravity.The gauge field introduces a mass scale and breaks parity in the gravitational wave system.A new symmetry also emerges among the gravitational wave and gauge field amplitudes.These effects may have a unique imprint on a spectrum of primordial gravitational waves,allowing us to gain unparalleled insight into the earliest moments in the history of theUniverse. Moreover, these results suggest that to go beyond linear order in the quantization6f gravity, the expanded set of gravitational degrees of freedom created by the backgroundmust be taken into account. 7 CKNOWLEDGMENTS
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