Vainshtein mechanism in massive gravity nonlinear sigma models 24th October 2016
Katsuki Aoki, Waseda University
Based on KA and Shuntaro Mizuno, PRD 94, 064054 (2016) [arXiv:1607.00173]
JGRG26, Osaka City University Introduction
What is dark matter? Ordinary Matter 4.9% What is dark energy? Dark Matter We may need beyond standard model!! 26.8% 68.3% Standard model of gravity = GR, but Dark Energy What is graviton? - It should be spin-2 field. - Massless field or Massive field?
The theory of massive graviton (Massive gravity or Bigravity) Massive graviton can explain dark energy or dark matter!! e.g., in bigravity, Volkov ’11, von Strauss et al. ’11, Cristosomi et al. ’11. for DE KA and Kei-ichi Maeda, ’14, KA and Shinji Mukohyama, ’16 for DM JGRG26, Osaka City University Local gravity constraint?
Gravity mediated by massive graviton
If the graviton mass is light enough → Yukawa potential may be restored into Newtonian potential in small scale? → consistent with experiments??
The linear theory of massive graviton (Fierz-Pauli theory) is not restored into GR!! van Dam and Veltman, (1970), Zakharov, (1970) GR could be recovered by Vainshtein mechanism. Vainshtein, (1970)
JGRG26, Osaka City University vDVZ discontinuity (Minkowski background)
Massive graviton = tensor modes + vector modes + scalar mode
Stueckelberg trick
General mass term (훼 ≠ −1)
→ Ostrogradsky ghost
Ghost-free (Fierz-Pauli) mass term (훼 = −1) → No higher derivative
JGRG26, Osaka City University vDVZ discontinuity (Minkowski background)
Ghost-free mass term (훼 = −1) → No higher derivative Scalar mode does not decouple from tensor mode. → GR is not recovered even in massless limit.
Nonlinear ghost-free mass terms
Galileon interaction
Perturbation can be used if → Perturbation theory cannot be used in massless limit
JGRG26, Osaka City University Vainshtein mechanism and decoupling limits
Vainshtein mechanism: Interactions are suppressed by nonlinear derivative interactions. and so on…
Energy scale of coupling effectively increases by Vainshtein mechanism.
The analysis of Vainshtein mechanism in bigravity (and massive gravity) is not so easy since there are many degrees of freedom. → The decoupling limit is useful to discuss the Vainshtein mechanism. Focusing on a scale, one can pick up only relevant interactions with keeping number of degrees of freedom.
JGRG26, Osaka City University 횲ퟑ decoupling limit → 횲ퟐ decoupling limit
Original relevant nonlinear interactions around Minkowksi background
→ Λ3 decoupling limit
However, tensor mode is not decoupled in Λ3 decoupling limit. Vainshtein mechanism leads to increasing of the energy scale of couplings. → Energy scale of relevant interactions is no longer Λ3. The relevant interactions have new scales
→ 휦ퟐ decoupling limit gives an effective theory inside Vainshtein radius
de Rham, Tolley, and Zhou, (2016). KA and Mizuno, (2016).
JGRG26, Osaka City University Effective theory inside Vainshtein radius
“Massive gravity nonlinear sigma model”
Metrics are solutions in GR since Vainshtein mechanism is implemented. The effective theory describes dynamics of Stueckelberg fields 흓풂. The number of degrees of freedom is three (vector + scalar modes)
de Rham, Tolley, and Zhou, (2016): 푔휇휈, 푓푎푏 = Minkowksi
KA and Mizuno, (2016): 푔휇휈 = curved by a source, 푓푎푏 = Minkowksi JGRG26, Osaka City University Instability of Vainshtein mechanism
Based on the effective theory, we have found that if the vector graviton is not excited, the scalar graviton has ghost or gradient instability in vacuum region. f-spacetime is assumed to be flat.
휋훼 is a vector on Minkowksi spacetime.
We consider the case when the vector graviton is not excited. g-spacetime is assumed to be a Ricci flat spacetime and gravity is weak. (= the space region far from the source)
JGRG26, Osaka City University Instability of Vainshtein mechanism
The scalar graviton is decomposed into background and perturbation.
(One can construct an explicit form of 휋ത in static and spherically symmetric case, but the result does not depend on the configuration of 휋ത)
The action of perturbation in weak gravity region
in Ricci flat spacetime
휇 휇휈 Signs of eigenvalues of 푍 휈 ≠ 1, 1, 1, 1 , i.e., 푍 ≠ −1, 1, 1, 1 → ghost or gradient instability in any Ricci flat spacetime
JGRG26, Osaka City University Summary and Discussions
The effective theory inside Vainshein radius → obtained from Λ2 decoupling limit. Effective theory reproduces known solutions of Vainshtein mechanism.
Based on the effective theory, we have found that if the vector graviton is not excited, the scalar graviton has ghost or gradient instability in vacuum region.
Successful Vainshein mechanism → Vector graviton is needed? (or massive graviton cannot be stably restored into massless graviton?)
*We have also explicitly calculated perturbations around the static and spherically symmetric solution, and found there indeed exists ghost or gradient instability.
JGRG26, Osaka City University