On the screening mechanism in DHOST theories evading gravitational wave constraints
RRUP-19-8
On the screening mechanism in DHOST theories evading gravitational waveconstraints
Shin’ichi Hirano, ∗ Tsutomu Kobayashi, † and Daisuke Yamauchi ‡ Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan Faculty of Engineering, Kanagawa University, Kanagawa, 221-8686, Japan
We consider a subclass of degenerate higher-order scalar-tensor (DHOST) theories in which grav-itational waves propagate at the speed of light and do not decay into scalar fluctuations. Thescreening mechanism in DHOST theories evading these two gravitational wave constraints operatesvery differently from that in generic DHOST theories. We derive a spherically symmetric solution inthe presence of nonrelativistic matter. General relativity is recovered in the vacuum exterior regionprovided that functions in the Lagrangian satisfy a certain condition, implying that fine-tuning isrequired. Gravity in the matter interior exhibits novel features: although the gravitational poten-tials still obey the standard inverse power law, the effective gravitational constant is different fromits exterior value, and the two metric potentials do not coincide. We discuss possible observationalconstraints on this subclass of DHOST theories, and argue that the tightest bound comes from theHulse-Taylor pulsar.
I. INTRODUCTION
Measuring the speed of gravitational waves serves asa powerful test for modified theories of gravity [1–4].Based on this idea, the nearly simultaneous detectionof the gravitational wave event GW170817 and its elec-tromagnetic counterpart GRB 170817A [5–7] was usedto put a tight limit on scalar-tensor theories as alterna-tives to dark energy [8–15]. Within the Horndeski classof scalar-tensor theories [17–19], derivative couplings ofthe scalar degree of freedom φ to the curvature have thusbeen ruled out. One can extend the Horndeski theory in ahealthy manner to degenerate higher-order scalar-tensor(DHOST) theories, in which Ostrogradsky instabilitiesare eliminated despite the higher-order Euler-Lagrangeequations [20–28] (see Refs. [29–31] for a review). Non-trivial derivative couplings to the curvature are still al-lowed in the context of DHOST theories. These theoriesare phenomenologically very interesting because whilethe Vainshtein screening mechanism is successfully imple-mented in the vacuum region exterior to matter distribu-tions, it is partially broken in the matter interior [32–35].This implies that DHOST theories can only be testedin the interior of extended objects such as stars, galaxyclusters, and Earth’s atmosphere [36–49].Recently, yet another constraint on DHOST theorieshas been pointed out: gravitons must be stable againstdecay into dark energy [50]. The Lagrangian for DHOSTtheories in which gravitons propagate at the speed of light ∗ Email: s.hirano”at”rikkyo.ac.jp † Email: tsutomu”at”rikkyo.ac.jp ‡ Email: yamauchi”at”jindai.jp The constraints have been imposed assuming that modified grav-ity as an alternative to dark energy is valid on much higher energyscales where LIGO observations are made, though this assump-tion may be subtle [16]. and do not decay into dark energy is described by L = G ( φ, X ) − G ( φ, X ) (cid:50) φ + f ( φ, X ) R + 3 f X f φ µ φ µσ φ σν φ ν , (1)where R is the Ricci scalar, φ µ = ∇ µ φ , φ µν = ∇ µ ∇ ν φ , X := − φ µ φ µ /
2, and f X = ∂f /∂X . Cosmology derivedfrom the Lagrangian (1) is explored in Ref. [51]. It turnsout that in this particular subclass of DHOST theoriesthe screening mechanism operates in a different way fromthat in generic DHOST theories, as already inferred inRef. [50]. The purpose of the present paper is to clarifyhow the (breaking of the) Vainshtein screening mecha-nism occurs in the above theory. II. SCREENING MECHANISM IN DHOSTTHEORIES WITHOUT GRAVITON DECAY
A weak gravitational field is described by the line ele-ment d s = − [1 + 2Φ( t, (cid:126)x )]d t + [1 − t, (cid:126)x )]d (cid:126)x , (2)with the scalar-field configuration φ = φ ( t ) + π ( t, (cid:126)x ) . (3)Here, φ ( t ) is a slowly evolving background determinedfrom the cosmological boundary condition and π ( t, (cid:126)x ) isa fluctuation. Since we are interested in gravity on scaleswell inside the horizon, we ignore the cosmic expansion.Following Refs. [32, 52], we expand the action in termsof the metric perturbations and π , keeping the higher-derivative terms relevant to the screening mechanismin the quasi-static regime. The resultant effective La- a r X i v : . [ g r- q c ] M a r grangian is given by L eff = f (cid:20) − ∂ Ψ + 4(1 − β )Ψ ∂ Φ − η f ( ∂π ) + 4 β (cid:18) − β (cid:19) Φ ∂ Φ + 4 ξf / Ψ ∂ π + 2( α − ξ ) f / Φ ∂ π + αf Λ ( ∂π ) ∂ π + 2 β (1 − β ) f / Λ ( ∂π ) ∂ Φ − βf / Λ ( ∂π ) ∂ Ψ+ 6 β f Λ ∂ i π∂ j π∂ i ∂ k π∂ k ∂ j π + 6 β f / Λ ( ∂ ˙ π ) − β (1 − β ) ˙ φ f / Λ Φ ∂ ˙ π + 8 β ˙ φ f / Λ Ψ ∂ ˙ π + 6 β ˙ φ f Λ ( ∂π ) ∂ ˙ π (cid:21) − Φ ρ, (4)where we introduced dimensionless quantities α := ˙ φ G X f / , β := ˙ φ f X f , ξ := f φ f / , (5)and defined an energy scale Λ := ( ˙ φ /f / ) / . The dotdenotes differentiation with respect to t . The explicit ex-pression for the coefficient η is not important here. Inderiving the Lagrangian (4) we ignored ¨ φ since φ isa slowly varying field. We assume that matter is mini-mally coupled to gravity, so that we add the term − Φ ρ where ρ = ρ ( t, (cid:126)x ) is the density of a nonrelativistic mat-ter source. The Lagrangian (4) is a particular case of thegeneral effective Lagrangian for the Vainshtein mecha-nism in DHOST theories [33–35]. However, the screeningmechanism in this particular subclass operates in a verydifferent way than in generic cases, as we will see below.Let us consider a spherically symmetric matter dis-tribution, ρ = ρ ( t, r ), where r is the radial coordinate.Varying the action with respect to Ψ, Φ, and π , we ob-tain the following equations:(1 − β ) ξx + (1 − β ) y − z − βx ( rx ) (cid:48) + 2 ˙ φ Λ β ˙ x = 0 , (6)[ α − ξ + (1 − β ) βξ ] x + 2 β (2 − β ) y + 2(1 − β ) z + 2 β (1 − β ) x ( rx ) (cid:48) − φ Λ β (1 − β ) ˙ x = A, (7)and F ( x, ˙ x, x (cid:48) , ¨ x, ˙ x (cid:48) , x (cid:48)(cid:48) , y, ˙ y, y (cid:48) , z, ˙ z, z (cid:48) ) = 0 , (8)where the prime denotes differentiation with respect to r and we defined the dimensionless variables as x := π (cid:48) Λ r , y := f / Φ (cid:48) Λ r , z := f / Ψ (cid:48) Λ r , (9) A := 18 π ˙ φ M ( t, r ) r = 18 πf / Λ M ( t, r ) r , (10) with M ( t, r ) := 4 π (cid:90) r ρ ( t, ¯ r )¯ r d¯ r (11)being the mass contained within r . In deriving Eqs. (6)–(8) we integrated the field equations once and fixed theintegration constants so that x , y , and z are regular at r = 0. The explicit form of F is complicated.From Eqs. (6) and (7) we have y = A + 2 β (1 − β ) x ( rx ) (cid:48) − β ) + c x − ˙ φ Λ β − β ˙ x, (12) z = (1 − β ) A − β (1 − β ) x ( rx ) (cid:48) − β ) + c x + ˙ φ Λ β − β ˙ x, (13)where c and c are written in terms of α , β , and ξ . Then,substituting Eqs. (12) and (13) to Eq. (8), we obtain4( α − βξ )(1 − β ) x + (cid:20) c − β (1 − β ) ( r A ) (cid:48) r (cid:21) x = [ α + (1 − β ) ξ − ζ ] A − φ Λ (1 − β ) β ˙ A, (14)where we defined ζ := ˙ φ f φX f / , (15)and the explicit expression for c (which is written interms of α , β , etc. and their time derivatives) is notimportant. As expected from the degeneracy of the the-ory, the final result (14) is just an algebraic equation for x , with no derivatives acting on x . In generic quadraticDHOST theories, however, one would obtain at this fi-nal stage a cubic equation for x . The present theory isspecial in the sense that the coefficient of the cubic termvanishes identically.From now on, let us consider the case where the sourceis static, ρ = ρ ( r ). Then, since we are assuming that ˙ φ is approximately constant, A is also independent of time.Thus, ˙ A in Eq. (14) can be neglected.One may define the typical radius r V below which non-linearities are large by A ( r V ) = 1. We are mainly inter-ested in the solutions to Eq. (14) for A (cid:29) A ∝ r − , whereas we have ( r A ) (cid:48) (cid:54) = 0inside.Let us first consider the exterior region. For A (cid:29) x (cid:39) ± (cid:20) α + (1 − β ) ξ − ζ ( α − βξ )(1 − β ) A (cid:21) / . (16)From this it can be seen that the terms linear in x inEqs. (12) and (13) are suppressed relative to the otherterms. We thus find, irrespective of the sign of Eq. (16),that y (cid:39) α (4 − β ) − β (13 − β ) ξ + 2 βζ α − βξ )(1 − β ) A, (17) z (cid:39) α (4 − β ) − β (1 − β ) ξ − βζ α − βξ )(1 − β ) A, (18)This shows that Φ (cid:54) = Ψ in general, implying that thepresent subclass of DHOST theories does not evade thesolar-system constraints. However, if the parameters sat-isfy α − ξ (1 + 10 β ) + 2 ζ = 0 , (19)general relativity is recovered, yielding y = z = A − β ) , ⇔ Φ (cid:48) = Ψ (cid:48) = 116 πf (1 − β ) Mr . (20)The effective gravitational constant is given by G N, out = 116 πf (1 − β ) . (21)Thus, fine-tuning is needed in order for the screen-ing mechanism to work successfully in the vicinity ofa source. This is in contrast to generic DHOST theo-ries [32–35].Next, let us look at the interior region. We have twobranches, one of which is given by(I) : x (cid:39) β α − βξ ) ( r A ) (cid:48) r (cid:29) , (22)and the other by(II) : x (cid:39) − α + (1 − β ) ξ − ζ β (1 − β ) r A ( r A ) (cid:48) = O (1) . (23)In Branch I, the behavior of gravity is far away fromthe normal one: y = 9 β (1 − β ) ξ ( r A ) (cid:48) r (cid:20) ( r A ) (cid:48)(cid:48) − ( r A ) (cid:48) r (cid:21) + O ( A ) , (24) z = − β (1 − β ) ξ ( r A ) (cid:48) r (cid:20) ( r A ) (cid:48)(cid:48) − ( r A ) (cid:48) r (cid:21) + O ( A ) , (25)where Eq. (19) was assumed. It then follows thatΦ (cid:48) (cid:39) − Ψ (cid:48) ∝ M (cid:48) M (cid:48)(cid:48) r − ( M (cid:48) ) r . (26) More precisely, the condition for successful screening is β [3 α − ξ (1 + 10 β ) + 2 ζ ] = 0. Clearly, the case with β = 0 correspondsto the subclass of the Horndeski theory. This is the trivial caseexhibiting the Vainshtein mechanism [52–54]. - ��� ���� ���� � �� ��������� � �� � �� � �� �� � FIG. 1. An example of a Branch II solution for r V = 1000and the stellar radius ∼
1. The dashed line corresponds tothe potentials in GR with the gravitational constant G N, out . We therefore conclude that this branch would not de-scribe the stellar structure appropriately, and hence mustbe excluded.Branch II is phenomenologically more interesting. Inthis branch, all x ’s in Eqs. (12) and (13) can be neglected,leading to y = A − β ) , z = (1 − β ) A − β ) . ⇔ Φ (cid:48) = 116 πf (1 − β ) Mr , Ψ = (1 − β )Φ . (27)From this we see that the effective gravitational constantinside the matter distribution is different from the exte-rior value by a factor of (1 − β ) − : G N, in = G N, out − β . (28)This must be contrasted with the way of breaking thescreening mechanism in generic DHOST theories, where M (cid:48) and M (cid:48)(cid:48) appear in Φ (cid:48) and Ψ (cid:48) as corrections to thestandard gravitational law with the same gravitationalconstant as the exterior one [32–35]. We also see that Φand Ψ do not coincide in the matter interior. One shouldnote that Eq. (19) is not used when deriving Eq. (27).Let us finally comment on the solution for A (cid:28)
1. Wehave two branches, namely, x ∼ y ∼ z ∼ A and x ∼ y ∼ z ∼
1. By inspecting the explicit solutions to Eq. (14), wefind that the former branch, which is phenomenologicallymore acceptable, is matched onto Branch II if β (1 − β ) c < x , y , and z for A ( r ) = B ( r ) /B (1000) (namely, r V =1000) with B ( r ) = ( r +1) − . The density profile mimicsa star with the radius r ∼
1. The parameters are givenby ξ = α = 1, β = ζ = 1 /
4, and c = 1. (For x we plotan exact solution to Eq. (14), but for y and z the terms - ��� ���� ���� � �� ��������� � �� � �� � �� �� � FIG. 2. The Branch II solution for the NFW density profile.The dashed line corresponds to the potentials in GR with thegravitational constant G N, out . linear in x are ignored because they are subdominant for r (cid:28) r V .)We also present in Fig. 2 the Branch II solution forthe NFW density profile, ρ ( r ) = ρ / [( r/r s )(1 + r/r s ) ]with r s = 1 and ρ chosen so that r V = 1000. Theparameters are again given by ξ = α = 1, β = ζ = 1 / c = 1. Since there is no definite surface in this case,we see deviations from general relativity everywhere. III. OBSERVATIONAL CONSTRAINTS
We have seen that though the particular subclass ofDHOST theories (1) could evade solar-system tests byrequiring the fine-tuned relation (19), (i) Φ and Ψ do notcoincide inside the matter distribution, and (ii) the grav-itational constant in the matter interior is different fromits exterior value. Let us discuss briefly possible obser-vational constraints on such modifications of gravity.The difference between the two potentials in the nonva-cuum region, Ψ / Φ − − β , can be measured by com-paring the X-ray and lensing profiles of galaxy clusters,as has been investigated for different types of modifica-tions in Refs. [41, 55, 56]. In particular, the constraintsobtained for beyond Horndeski theories in Ref. [41] read | Φ / Φ GR − | < O (10 − ) and | Ψ / Ψ GR − | < O (10 − ).Thus, we would expect constraints of the same order ofmagnitude, | β | < O (10 − ), from galaxy clusters.A different value of the gravitational constant insidethe Sun would lead to changes in the solar structure, andthereby modify the sound speed and solar neutrino fluxes.Based on the solar standard model, it has been arguedthat a relative difference of O (10 − ) is still allowed byobservations [57]. Thus, the Sun could potentially beused to test a different value of the gravitational constantinside extended objects.Note, however, that currently the most stringentbound comes from the difference between the measured value of the gravitational constant, G N (= G N, out or G N, in ), and the gravitational coupling for gravitationalwaves, G GW , which is constrained from the orbital decayof the Hulse-Taylor pulsar: − . × − < G GW /G N − < . × − [35, 58]. In the present case, we have G GW = (16 πf ) − [28, 59], so the constraint is given by | β | < O (10 − ) , (30)which is orders of magnitude tighter than the possibleconstraint from galaxy clusters. IV. CONCLUSIONS
In this paper, we have studied the screening mechanismin a particular subclass of degenerate higher-order scalar-tensor (DHOST) theories in which the speed of gravita-tional waves is equal to the speed of light and gravitonsdo not decay into scalar fluctuations. By inspecting aspherically symmetric gravitational field, we have foundthat the screening mechanism operates in a very differentway from that in generic DHOST theories [32–35]. First,the fine-tuning is required so that solar-system tests areevaded in the vacuum exterior region. This is in contrastto generic DHOST theories, in which the implementationof the Vainshtein screening mechanism outside the mat-ter distribution is rather automatic. Second, the way ofthe Vainshtein breaking inside extended objects is alsodifferent from that in generic DHOST theories. We haveshown that in the interior region the metric potentialsobey the standard inverse power law, but the two do notcoincide. Moreover, the effective gravitational constantdiffers from its exterior value. However, the current moststringent bound comes from the fact that the effectivegravitational coupling for gravitational waves is differentfrom the Newtonian constant [35, 58], rather than fromthe above interesting phenomenology. The obtained con-straint is as tight as (cid:12)(cid:12)(cid:12)(cid:12) Xf X f (cid:12)(cid:12)(cid:12)(cid:12) < O (10 − ) . (31)Thus, we conclude that the allowed parameter space issmall for DHOST theories as alternatives to dark energyevading gravitational wave constraints. ACKNOWLEDGMENTS
The work of SH was supported by the JSPS Re-search Fellowships for Young Scientists No. 17J04865.The work of TK was supported by MEXT KAKENHIGrant Nos. JP15H05888, JP17H06359, JP16K17707,JP18H04355, and MEXT-Supported Program for theStrategic Research Foundation at Private Universities,2014-2018 (S1411024). The work of DY was supportedby MEXT KAKENHI Grant No. JP17K14304. [1] A. Nishizawa and T. Nakamura,
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