Determining Cosmological Constant Using Gravitational Wave Information

It is shown in Einstein gravity that the cosmological constant Lambda introduces a graviton mass m into the theory, a result that will be derived from the Regge-Wheeler-Zerilli problem for a particle falling onto a Kottler-Schwarzschild mass with Lambda. The value of m is precisely the Spin-2 gauge line appearing on the Lambda versus m2 phase diagram for Spin-2, the partially massless gauge lines introduced by Deser&Waldron and described as the Higuchi bound. Note that this graviton is unitary with only four polarization degrees of freedom (helicities plus&minus 2 and 1, but not 0 because a scalar gauge symmetry removes it). The conclusion is drawn that Einstein gravity (with Lambda) is a partially massless gravitation theory which has lost its helicity 0 due to a scalar gauge symmetry. That poses a challenge for gravitational wave antennas as to whether they can measure the loss of this gauge symmetry. Also, given the recent results measuring the Hubble constant Ho from LIGO-Virgo data, it is then shown that Lambda can be determined from the LIGO results for the graviton mass m and Ho. This is yet another multi-messenger source for determining the three parameters Lambda, m, and Ho in astrophysics and cosmology, at a time when there is much disparity in measurements of Ho.

It is shown in Einstein gravity that the cosmological constant Λ introduces a graviton mass m g into the theory, a result that will be derived from the Regge-Wheeler-Zerilli problem for a particle falling onto a Kottler-Schwarzschild mass with Λ≠0. The value of m g is precisely the Spin-2 gauge line appearing on the Λ-m g 2 phase diagram for Spin-2, the partially massless gauge lines introduced by Deser & Waldron in the (m g 2 , Λ) phase plane and described as the Higuchi bound m g 2 = 2Λ/3. Note that this graviton is unitary with only four polarization degrees of freedom (helicities ±2, ±1, but not 0 because a scalar gauge symmetry removes it). The conclusion is drawn that Einstein gravity (EG, Λ≠0) is a partially massless gravitation theory. Given the recent results measuring the Hubble constant H o from LIGO-Virgo data, it is then shown that Λ can be determined from the LIGO results for the graviton mass m g and H o . This is yet another multi-messenger source for determining the three parameters Λ, m g , and H o in astrophysics and cosmology. Introduction: In order to determine the graviton mass of Einstein gravity (EG), we proceed as follows. A curved Kottler-Schwarzschild (KS) metric with Λ≠0 will be applied to the Regge-Wheeler-Zerilli (RWZ) problem [1][2][3][4][5] representing gravitational radiation perturbations produced by a particle falling onto a large mass M. The RWZ result (Λ=0) will be extended to the general EG problem with Λ≠0 (EGΛ), in the fashion that Kottler extended the Schwarzschild metric to de Sitter space (SdS).
One begins with a small perturbative expansion of the Einstein field equations R μν -½g μν R + Λg μν = -κT μν (1) about the known exact solution η μν where the metric tensor is g μν = η μν + h μν , with h μν the dynamic perturbation of the background raising and lowering operator η μν . The most general spherically symmetric solution is well-known to be a Kottler-Schwarzschild (KS) metric ds 2 =e ν dt 2 + e ζ dr 2 + r 2 dΩ 2 , where with M = GM*/c 2 , dΩ 2 = (dΘ + sin 2 dϕ 2 ), and η μν = diag(e ν ,e -ν ,r 2 ,r 2 sin2Θ) in spherically symmetric coordinates. Its contravariant inverse η μν is defined such that η μν η μν = δ μ ν . The wave equation for gravitational radiation h μν on the non-flat background containing Λ in (1) will follow as (9) below, derived now from the procedure developed in the RWZ formalism. Perturbation analysis of (1) for a stable background η μν = g (0) μν produces the following ; ; Stability must be assumed in order that δT μν is small. This equation can be simplified by defining the function (introduced by Einstein himself) Substituting (5) and (6) into (4) and re-grouping terms gives Now impose the Hilbert-Einstein-de-Donder gauge which sets (6) to zero (f μ = 0), and suppresses any vector gravitons. Wave equation (7) reduces to In an empty ( which is the starting point for the RWZ formalism. Weak-Field Limit, de Sitter Metric. The Schwarzschild character of the RWZ problem above will now be relaxed, with η μν again diagonal, but M = 0 and Λ≠0 in (2) and (3). The wave equation of paramount importance will follow as (17).
We know that the trace of the field equations (1) gives 4Λ − R = − κT, whereby they become For an empty space (T μν = 0 and T = 0), (10) reduces to de Sitter space and the trace to R = 4Λ. Substitution of R and R μν from (11) into (8) using (5) shows that the contributions due to Λ ≠ 0 are of second order in h μν . Neglecting these terms (particularly if Λ is very, very small) simplifies (8) to One can arrive at (12) to first order in h μν by using g μν as a raising and lowering operator rather than the background η μν − a result which incorrectly leads some to the conclusion that Λ terms cancel in the gravitational wave equation. Note with caution that (12) and the RWZ equation (9) are not the same wave equation. Overtly, the cosmological terms have vanished from (12), just like (9) where Λ was assumed in the RWZ problem to be nonexistent in the first place. However, the character of the Riemann tensor R α μν β is significantly different in these two relations.
Simplifying the SdS metric by setting the central mass M * in η μν to zero, produces the de Sitter space (11) of constant curvature K = 1/R 2 , where we can focus on the effect of Λ. The Riemann tensor is now and reverts to for use in (12). This substitution (raising and lowering with η μν ) into (12) next gives K and Λ term contributions to second order in h μν . Recalling that curvature K is related to Λ by K = Λ/3, substitution of (15) back into (12) gives to first order There is no cancellation of the Λ contributions to first order. Noting from (5) (17) for the limit r→0. From (17) and (18) (19) in the locally flat-space limit r<<1.
Note that Penrose [6] has pointed out that due to conformal invariance arguments, the massless Klein-Gordon equation becomes (□ -R/6)φ = 0 on a curved background. This necessarily gives (18) since R=4Λ in de Sitter space. Also in passing, by rescaling h as 2 h → ½ 1 h in (12) and (17), then (18) which is the surface gravity κ C = m g of the cosmological event horizon identified by Gibbons & Hawking [7]. It is also found in Weinberg [8].
Locally Flat Limit of Wave Equation (17). It is necessary to demonstrate that hidden Λ-terms arising from To simplify calculations, now note that r 2 dΩ 2 in (2) is of second-order in r and is negligible as r→0. Thus the focus is on e ν (with M=0) in (3) appearing in the diagonal of η μν and its inverse η μν . Hence, η 00 =−c and η 00 =−c -1 , while η 11 =c -1 and η 11 = c. Also, note that c(r)→1 and c(r) -1 →1 as r→0. Introducing where □ B μν is the term of interest. A μν and C μν contain factors of second order, or terms that vanish in locally flat space (r<<1). Furthermore, only the first-order second derivatives in B μν remain as r→0. These terms are In this approximation, □= − ∂ t 2 + 2  → 2  . Also □η 00 → 2  η 00 = +⅔λ and □η 11 → 2  η 11 = +⅔λ. We find that h h □  h in the locally flat limit of (17). The graviton mass (18) for EGΛ thus follows from this analysis, a result first determined many years ago [9].
In Conclusion. These results come directly from the RWZ equation (9). The consequence is yet another way to determine the cosmological constant Λ, but from gravitational wave observations. It constitutes an entirely new prediction from Einstein's theory, that Λ, c, H o , and m g (having only 4 Spin-2 DOFs with helicities ±2, ±1), and conventional Λ-lore such as dark matter in ΛCDM models, are interrelated. For that reason alone, (18) needs to be verified experimentally. In addition, all of these parameters must collectively produce self-consistent values. The answer may also contribute to our understanding of galacticrotation-curve behavior and the accelerating Universe should the much-discussed Yukawa-potential implications of an m g like (18) prove to be true. Such predictions by EGΛ need to be investigated further. The fundamental question for partially massive gravity is whether existing gravitational wave antenna configurations can be used to measure or determine the loss of the helicity 0 polarization caused by loss of a scalar gauge symmetry. It will probably require additional antenna configurations and possibly more antennas. ________________