(Cosmology), black holes and stars in beyond Horndeski theoryjuryokuha/2016_Kyoto_babichev.pdf · (Cosmology), black holes and stars in beyond Horndeski theory Eugeny Babichev LPT

(Cosmology), black holes and stars

in beyond Horndeski theory

Eugeny Babichev

LPT Orsay

Black holes and stars in modified

gravity

Eugeny Babichev

A Thesis submitted for the degree of Habilitation a Diriger desRecherches

Laboratoire de Physique Theorique, CNRS, Univ. Paris-Sud,Universite Paris-Saclay, 91405 Orsay, France

August 2016

Black holes and stars in modified

gravity

Eugeny Babichev

A Thesis submitted for the degree of Habilitation a Diriger desRecherches

Laboratoire de Physique Theorique, CNRS, Univ. Paris-Sud,Universite Paris-Saclay, 91405 Orsay, France

August 2016

GWastro Symposium 2016, YITP

collaboration with C. Charmousis, G. Esposito-Farese, K. Koyama, D. Langlois, R. Saito, J. Sakstein

Many ways to modify gravity:!❖ Scalar-tensor theories: Brans-Dicke, f(R), (Genralized) Galileons,

(beyond) Horndeski theory, KGB, Fab-four…!❖ Higher-dimensions!❖ Horava, Khronometric theory!❖ Massive and bi-gravity

Modified gravity?

Why modification of gravity?!❖ Cosmological constant problems!❖ Dark matter!❖ Non-renormalizability problem!❖ Theoretical curiosity !❖ Benchmarks for testing General Relativity

See Shinji’s talk for introduction

Plan

❖ Motivation!

❖ Beyond Horndeski (generalised Galileons)!

❖ Cosmology and self-tuning of CC!

❖ Local physics!

❖ Black holes!

❖ Neutron stars

❖ Schwarzschild-de-Sitter solution:!

❖ Note that !

❖ Solution self tunes vacuum cosmological constant. “Action induced" effective cosmological constant appears.!

❖ NB.

some motivation

S =

Zd

4x

p�g

h⇣R� 2⇤� ⌘ (@�)2 + �G

µ⌫@µ�@⌫�

i

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f = h = 1� µ

r+

⌘

3�r2

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⇤e↵ = �⇣⌘/�<latexit sha1_base64="EA9eQr0VGWYYs+YMWqwx1cZ2Elw=">AAACDnicbVC7SgNBFJ2Nrxhfq5Y2g0GxMW5E1EIhYGNhEcGYQDaE2cndZMjsg5m7YlzyBzb+io2Fiq21nX/j5FFo4oW5HM45lzv3eLEUGh3n28rMzM7NL2QXc0vLK6tr9vrGrY4SxaHCIxmpmsc0SBFCBQVKqMUKWOBJqHrdi4FevQOlRRTeYC+GRsDaofAFZ2iopr3rXhlzizVdhHtMwff79Jzuuw+AjLqmHbie6U077xScYdFpUByDPBlXuWl/ua2IJwGEyCXTul50YmykTKHgEvo5N9EQM95lbagbGLIAdCMd3tOnO4ZpUT9S5oVIh+zviZQFWvcCzzgDhh09qQ3I/7R6gv5pIxVhnCCEfLTITyTFiA7CoS2hgKPsGcC4EuavlHeYYhxNhDkTQnHy5GlQOSwcF5zro3zpbJxGlmyRbbJHiuSElMglKZMK4eSRPJNX8mY9WS/Wu/Uxsmas8cwm+VPW5w9r6JvO</latexit><latexit sha1_base64="EA9eQr0VGWYYs+YMWqwx1cZ2Elw=">AAACDnicbVC7SgNBFJ2Nrxhfq5Y2g0GxMW5E1EIhYGNhEcGYQDaE2cndZMjsg5m7YlzyBzb+io2Fiq21nX/j5FFo4oW5HM45lzv3eLEUGh3n28rMzM7NL2QXc0vLK6tr9vrGrY4SxaHCIxmpmsc0SBFCBQVKqMUKWOBJqHrdi4FevQOlRRTeYC+GRsDaofAFZ2iopr3rXhlzizVdhHtMwff79Jzuuw+AjLqmHbie6U077xScYdFpUByDPBlXuWl/ua2IJwGEyCXTul50YmykTKHgEvo5N9EQM95lbagbGLIAdCMd3tOnO4ZpUT9S5oVIh+zviZQFWvcCzzgDhh09qQ3I/7R6gv5pIxVhnCCEfLTITyTFiA7CoS2hgKPsGcC4EuavlHeYYhxNhDkTQnHy5GlQOSwcF5zro3zpbJxGlmyRbbJHiuSElMglKZMK4eSRPJNX8mY9WS/Wu/Uxsmas8cwm+VPW5w9r6JvO</latexit>

(@�)2 = const

�(t, r) = q t± q

h

p1� h

EB& Charmousis’13

Big CC problem

bare G ≡ 8π/M2Pl we introduce in this action, but it acquires a renormalized value Geff. Our

notation MPl and G should thus be understood as bare parameters, whose numerical valuesare not known yet. We will relate them to the observed ones in Sec. V, for a specific class ofmodels which reproduces the Schwarzschild solution in the vicinity of a spherical body. Inorder not to introduce extra hidden scales in the model, we will assume that all functions ofX defined below involve dimensionless coefficients of order O(1).

The class of theories we are considering is thus defined by the full action

S =M2

Pl

2

! √−g (R − 2Λbare) d

4x+"

(n,p)

! √−g L(n,p)d

4x+ Smatter[ψ, gµν ], (3)

where all matter fields different from ϕ (globally denoted as ψ) are assumed to be universallycoupled to gµν but not directly to ϕ, and where the generalized Horndeski Lagrangians L(n,p)

are related to the generalized Galileon ones (1) by

L(2,0) = M2f2(X)L(2,0) = −M4Xf2(X), (4a)

L(3,0) = f3(X)L(3,0), (4b)

L(4,0) =1

M2f4(X)L(4,0), (4c)

L(5,0) =1

M4f5(X)L(5,0), (4d)

L(4,1) = s4(X)L(4,1), (4e)

L(5,1) =1

M2s5(X)L(5,1). (4f)

Since different notation is used in the literature to define these theories, let us give adictionary. First of all, let us recall that the L(4,1) and L(5,1) of Refs. [7, 13] were not definedas in Eqs. (1g) and (1h) above, but rather as L(4,1) and L(5,1), Eqs. (4e) and (4f), with s4 =s5 = −X andM = 1. Second, generalized Horndeski theories were first defined in [10, 11, 14]with a notation mixing the Gn (ϕ2

λ) functions used for the Horndeski theory [4, 8, 34] andnew functions Fn (ϕ2

λ) multiplying the above contractions (1c) and (1e) with two Levi-Civitatensors:

L(2,0) = G2

#

ϕ2λ

$

, (5a)

L(3,0) = G3

#

ϕ2λ

$

!ϕ + tot. div., (5b)

L(4,0) + L(4,1) = G4

#

ϕ2λ

$

R− 2G′4

#

ϕ2λ

$

%

(!ϕ)2 − ϕµνϕµν&

+F4

#

ϕ2λ

$

εµνρσ εαβγσ ϕµ ϕα ϕνβ ϕργ + tot. div., (5c)

L(5,0) + L(5,1) = G5

#

ϕ2λ

$

Gµνϕµν +1

3G′

5

#

ϕ2λ

$

%

(!ϕ)3 − 3!ϕϕµνϕµν + 2ϕµνϕ

νρϕ µρ

&

+F5

#

ϕ2λ

$

εµνρσ εαβγδ ϕµ ϕα ϕνβ ϕργ ϕσδ + tot. div., (5d)

where G′4 (ϕ

2λ) and G′

5 (ϕ2λ) mean the derivatives of these functions with respect to their

argument, i.e., G′n (ϕ

2λ) = dGn (ϕ2

λ) /d (ϕ2λ) = dGn(−M2X)/d(−M2X). The partial integra-

tions given in Appendix A below imply that these functions Gn and Fn are related to our

6

Zero-point energy of quantum fields!❖ The zero-point energy formally diverges, as !❖ Naive “hard” cutoff gives !❖ At the same time the observed ( ) !❖ Hard cutoff is not Lorentz invariant, one should use other

regularisation schemes. Indeed, dimensional regularisation gives a different number

k ! 1⇢ ⇠ M4

Pl

⇢obs

⇠ 10�122M4

Pl

|⇢| ⇠ 1054⇢obs

Phase transitions!❖ Electroweak symmetry breaking !❖ QCD phase transition:

|⇢EW| ⇠ 108 GeV4

|⇢QCD| ⇠ 10�2 GeV4

I. INTRODUCTION

The huge discrepancy of the observed value of the cosmological constant and its varioustheoretical predictions is a long standing problem of modern physics. The value of theenergy density corresponding to the cosmological constant today, as fitted by observationsusing the ΛCDM model, is of order 10−46GeV4. This value, written in units of the Planckmass (MPl) is ∼ 10−122, which is to be compared to the naive theoretical prediction ofthe vacuum energy of order of Planck energy density. In other words, the naive predictedvalue of the vacuum energy density is 10122 times greater than the observed one. Thetheoretical estimate of the value of the vacuum energy comes from the existence of a zero-point energy of the quantized fields. The zero-point energy density formally diverges, asit contains an integral over all momenta of a given energy in each mode. However, theapplication of a cutoff at the Planck mass gives a vacuum energy density ρ ∼ M4

Pl. It hasbeen argued, however, that one should use a different regularization scheme, which does notbreak Lorentz invariance, see the review [1]. Dimensional regularization gives in particular adifferent answer, |ρ| ∼ 108GeV4 [2]. The problem is clearly alleviated, but the discrepancyremains nevertheless huge, i.e., the value of the vacuum energy density predicted in thisscheme is ∼ 1054 times greater than the observed one.

Besides the above mentioned problem of zero-point energy of quantum fluctuations, thereis yet another source of a big cosmological constant: phase transitions in the early Universe.In particular, the electroweak symmetry breaking, through which the gauge bosons gain theirmasses, is accompanied with a change of the vacuum value of the Higgs boson. This leads, inturn, to a change of vacuum energy density, which is estimated to be |ρEW| ∼ 108GeV4 [1].Similar phase transition in QCD physics leads to |ρQCD| ∼ 10−2GeV4 [3]. Any of thesepredictions leads to too large vacuum energy.

Modifying gravity by the introduction of a scalar degree of freedom in the gravity sec-tor is a promising attempt to solve the cosmological constant problem. The most generalscalar-tensor theory with equations of motion up to second order in derivatives is knownas the Horndeski theory [4], or, in modern formulations, the Galileons [5–8]. The absenceof higher than second derivatives in the equations of motion guarantees the absence of anyOstrogradski ghost — an extra ghost degree of freedom generically associated with higherderivatives. The opposite is not always true, however, i.e., equations of motion involvinghigher-order derivatives do not necessary imply the appearance of an extra degree of freedom.An extension of the Horndeski theory has indeed been constructed, “beyond Horndeski” the-ory [9–13], which leads to third-order equations of motion, but nevertheless with only onescalar degree of freedom.1

It has been shown that a subclass of the Horndeski/Galileon theory, called “Fab Four”, hasthe property of total cancellation of a bare cosmological constant [16, 17]. An extension of theFab Four model, which includes the beyond Horndeski terms holds the same property [18].In these scenarii the metric is flat, while the scalar field has a non-trivial configuration.Therefore this particular model cannot be realistic, since the observed Universe contains asmall but non-zero cosmological constant. One should thus search for a model which wouldbe able to self-tune, i.e., to naturally tune the large value of a bare cosmological constant toa small observed one. An example of such a model, in a peculiar non-linear extension of a

1 A further extension of the beyond Horndeski theory has also been studied in [14, 15]. We however do not

consider this “beyond beyond Horndeski” extension in the present paper.

2

Galileons/Horndeski theory

A(uxx

uyy

� u2xy

) + Buxx

+ Cuxy

+Duyy

+ E = 0

Monge-Ampère equation

- to find a surface with a prescribed Gaussian curvature !- optimizing transportation costs

The most generic scalar-tensor theory in 4D, whose equations of motion contain no more than second derivatives !(no Ostrogradski ghost)

“Universal field equations”

Horndeski‘1974

Monge‘1784, Ampère‘1820

Fairlie et al‘1991

S =

Zd

4xF

⇥g, @g, @

2g,', @', @

2'

⇤E[g, @g, @2g,', @', @2'] = 0

❖ Non-Covariant Galileon: scalar field with theory with “galileon” symmetry. Equations of motion are strictly of the second order and metric is non-dynamical!

❖ Covariant Galileon: adding non-minimal scalar-matter coupling to flat Galileon.!

Galileons/Horndeski theory

[Deffayet et al’09] + many other works

Shift-symmetric version: S =

Zd

4x

p�g

5X

i=2

Li

[Nicolis et al’08]

❖ The Horndeski theory gives second order EOMs => no Ostrogradski ghost!

❖ The inverse is not true: EOMs may contain higher order derivatives without adding an extra d.o.f.!

❖ Beyond Horndeski theory, +2 free functions

Beyond Horndeski

[Gleyzes et al’14, Zumalacarregui&Garcia-Bellido’13 ]

Beyond Horndeski




S =M2

Pl

2


4x+"

(n,p)

! √−g L(n,p)d




L(2,0) = M2f2(X)L(2,0) = −M4Xf2(X), (4a)

L(3,0) = f3(X)L(3,0), (4b)

L(4,0) =1

M2f4(X)L(4,0), (4c)

L(5,0) =1

M4f5(X)L(5,0), (4d)

L(4,1) = s4(X)L(4,1), (4e)

L(5,1) =1

M2s5(X)L(5,1). (4f)




L(2,0) = G2

#

ϕ2λ

$

, (5a)

L(3,0) = G3

#

ϕ2λ

$


L(4,0) + L(4,1) = G4

#

ϕ2λ

$

R− 2G′4

#

ϕ2λ

$

%


+F4

#

ϕ2λ

$


L(5,0) + L(5,1) = G5

#

ϕ2λ

$

Gµνϕµν +1

3G′

5

#

ϕ2λ

$

%


νρϕ µρ

&

+F5

#

ϕ2λ

$


where G′4 (ϕ

2λ) and G′



2λ) = dGn (ϕ2



6




S =M2

Pl

2


4x+"

(n,p)

! √−g L(n,p)d




L(2,0) = M2f2(X)L(2,0) = −M4Xf2(X), (4a)

L(3,0) = f3(X)L(3,0), (4b)

L(4,0) =1

M2f4(X)L(4,0), (4c)

L(5,0) =1

M4f5(X)L(5,0), (4d)

L(4,1) = s4(X)L(4,1), (4e)

L(5,1) =1

M2s5(X)L(5,1). (4f)




L(2,0) = G2

#

ϕ2λ

$

, (5a)

L(3,0) = G3

#

ϕ2λ

$


L(4,0) + L(4,1) = G4

#

ϕ2λ

$

R− 2G′4

#

ϕ2λ

$

%


+F4

#

ϕ2λ

$


L(5,0) + L(5,1) = G5

#

ϕ2λ

$

Gµνϕµν +1

3G′

5

#

ϕ2λ

$

%


νρϕ µρ

&

+F5

#

ϕ2λ

$


where G′4 (ϕ

2λ) and G′



2λ) = dGn (ϕ2



6

covariant derivative. The six generalized Galileon Lagrangians read

L(2,0) ≡ −1

3!εµνρσ εανρσ ϕµ ϕα = (∂µϕ)

2, (1a)

L(3,0) ≡ −1

2!εµνρσ εαβρσ ϕµ ϕα ϕνβ = (ϕµ)

2!ϕ− ϕµϕµνϕ

ν , (1b)

L(4,0) ≡ −εµνρσ εαβγσ ϕµ ϕα ϕνβ ϕργ (1c)

= (ϕµ)2 (!ϕ)2 − 2ϕµϕµνϕ

ν!ϕ− (ϕµ)

2(ϕνρ)2 + 2ϕµϕµνϕ

νρϕρ, (1d)

L(5,0) ≡ −εµνρσ εαβγδ ϕµ ϕα ϕνβ ϕργ ϕσδ (1e)

= (ϕµ)2 (!ϕ)3 − 3 (ϕµϕµνϕ

ν) (!ϕ)2 − 3(ϕµ)2(ϕνρ)

2!ϕ

+6ϕµϕµνϕνρϕρ !ϕ + 2(ϕµ)

2ϕ ρν ϕ

σρ ϕ

νσ

+3 (ϕµν)2 ϕρϕρσϕ

σ − 6ϕµϕµνϕνρϕρσϕ

σ, (1f)

L(4,1) ≡ −εµνρσ εαβγσ ϕµ ϕαRνρβγ = −4Gµνϕµϕν , (1g)

L(5,1) ≡ −εµνρσ εαβγδ ϕµ ϕα ϕνβ Rρσγδ (1h)

= 2(ϕµ)2R!ϕ− 2ϕµϕµνϕ

ν R− 4ϕµRµνϕν!ϕ

−4(ϕµ)2ϕνρRνρ + 8ϕµϕµνR

νρϕρ + 4ϕµϕνϕρσRµρνσ. (1i)

These definitions coincide with those of [7, 13] for all L(n,0). For those involving one Riemanntensor, L(n,1), we decided to simplify them by removing a factor (ϕλ)2. We shall indeedmultiply below all these Lagrangians by arbitrary functions of (ϕλ)2, therefore this extrafactor was unnecessarily heavy in definitions (1g) and (1h). Note that when multiplying theabove Lagrangians (1g) and (1h) by arbitrary functions of ϕ, L(4,1) was nicknamed “John”in the “Fab Four” model [16, 17], while L(5,1) was nicknamed “Paul”.

Generalized Horndeski theories correspond to multiplying the above Lagrangians by ar-bitrary functions of both the scalar field ϕ and its standard kinetic term (ϕλ)2. [We shallrecall the difference between Horndeski and generalized Horndeski theories below Eqs. (6).]In the present paper, we will focus on shift-symmetric theories, whose actions do not involveany undifferentiated ϕ, and we shall thus only multiply the above Lagrangians by functionsof (ϕλ)2.

In the following, we choose that ϕ is dimensionless, but introduce a mass scale M so thatall Lagrangians have the same dimension. The functions will depend on the dimensionlessratio

X ≡−(ϕλ)2

M2. (2)

Note the sign, the absence of a factor 12 , and the 1/M2 factor, as the notation X is used

with various definitions in the literature. Our negative sign is chosen so that X > 0 incosmological situations, where the time derivative ϕ of the scalar field is dominating overits spatial derivatives.

In addition to the mass scale M , which will be the only one we use in the scalar fieldkinetic terms, the action we consider also depends on two other scales: the reduced Planckmass MPl ≡ (8πG)−1/2 (in units such that ! = c = 1), which multiplies the Einstein-Hilbert action, and a bare cosmological constant Λbare, which may be much larger thanthe observed one (see Sec. III below for a discussion of the effective cosmological constantΛeff which is actually observed). A simple framework would be for instance to assume thatΛbare = O (M2

Pl), so that the model depend only on two scales, M and MPl. Let us stressthat the measured Newton’s constant, for instance in Cavendish experiments, is not the

5

Beyond Horndeski action:

Cosmology

Equations of motion

Gµ⌫ + ⇤bare gµ⌫ =Tµ⌫

M2Pl

rµJµ = 0

metric equations

scalar equation

Tµ⌫ ⌘ 2p�g

�S[']

�gµ⌫

Jµ ⌘ �1p�g

�S[']

�(@µ')

A consequence of the conserved!Noether current due to the shift-symmetry

Cosmology

first derivative of the expression within the curly brackets, because of the full antisymmetryof the Levi-Civita tensors. Therefore, although they contain in general first and even secondderivatives of the functions s4 and s5, they do not generate any s′′ in the divergence of (10).The conclusion is that the above combination (10) of the Einstein equations with the scalarcurrent cancels most of the first derivatives of the functions fn(X) and sn(X), but not allof them. We will use it in Secs. III and IV below, and we will see that such first derivativesactually do cancel in two important cases. In the homogenous and isotropic case of Sec. III,the reason is that we will focus on the time-time component of the Einstein equations, i.e.,µ = ν = 0 in Eq. (12). But in order to create first derivatives of s4 or s5, at least oneof the covariant derivatives ∇ρ or ∇σ must act on these functions, therefore at least oneamong ϕα or ϕβ must not be differentiated any more, and should thus correspond to theonly nonvanishing component ϕ0 = ϕ in this cosmological background. In other words, onemust have α = 0 or/and β = 0 to create a derivative of s in Eq. (12), and there will thusbe two indices 0 contracted with the same antisymmetric tensor ε, either µ = α = 0 or/andν = β = 0. This explains why (12) will not contribute to T 00 in Sec. III below, and whyall derivatives of fn(X) or sn(X) will be canceled in the combination (10). In the staticand spherically symmetric case of Sec. IV, we will see that some s′4 and s′5 do remain, butthey cancel in the particular case X = const. that we will study (and they actually cancelas soon as one assumes X = const., independently of spherical symmetry). It is indeedclear that (12) does not contribute to any derivative of s4 nor s5 if X = const. It is alsoeasy to prove that the only derivatives of functions f or s entering T µν are of the form2(ϕµϕν/M2)f ′(X)L(n,p) when X = const. (or with s′ instead of f ′), because they come fromthe variation of the metric used in the contraction X = −gµνϕµϕν/M2. On the other hand,the scalar current Jµ contains 2(ϕµ/M2)f ′(X)L(n,p), and no other derivative of a functionwhen we assume X = const. Therefore, the linear combination (10) obviously cancels thefew possible f ′ or s′ which occur when X = const.

III. COSMOLOGICAL SELF-TUNING

We consider a homogeneous and isotropic Universe whose metric takes the Friedmann-Lemaıtre-Robertson-Walker (FLRW) form

ds2 = −dt2 + a2(t)

!

dr2

1− kr2+ r2

"

dθ2 + sin2 θ dφ2#

$

, (13)

the parameter k ∈ {−1, 0, 1} determining whether the spatial hypersurfaces are open, flator closed, and we assume consistently that the scalar field ϕ depends only on t. Then itscurrent equation (9b) simply reads ∇0J0 = 0 ⇒ ∂t(a3J0) = 0, and its solution is thusJ0 = C0/a3, where C0 is a constant. This integration constant may be neglected at lateenough times, when the scale factor a becomes very large, and we will thus just solve forJ0 = 0 in the following (keeping in mind that an extra C0/a3 may be added to it).

Once the matter field equations are taken into account, i.e., ∇µJµ = 0 in the presentcase, it is well known that only the time-time-component of the Einstein equations (9a)needs to be solved. Indeed, the covariant conservation of the Einstein tensor ∇µGµν = 0

implies Gij = −gij%

G00 + G00/(3H)&

, where a dot denotes time differentiation andH ≡ a/a,

therefore the spatial components of the Einstein tensor are automatically solved once G00 is,

9

' = '(t)

first derivative of the expression within the curly brackets, because of the full antisymmetryof the Levi-Civita tensors. Therefore, although they contain in general first and even secondderivatives of the functions s4 and s5, they do not generate any s′′ in the divergence of (10).The conclusion is that the above combination (10) of the Einstein equations with the scalarcurrent cancels most of the first derivatives of the functions fn(X) and sn(X), but not allof them. We will use it in Secs. III and IV below, and we will see that such first derivativesactually do cancel in two important cases. In the homogenous and isotropic case of Sec. III,the reason is that we will focus on the time-time component of the Einstein equations, i.e.,µ = ν = 0 in Eq. (12). But in order to create first derivatives of s4 or s5, at least oneof the covariant derivatives ∇ρ or ∇σ must act on these functions, therefore at least oneamong ϕα or ϕβ must not be differentiated any more, and should thus correspond to theonly nonvanishing component ϕ0 = ϕ in this cosmological background. In other words, onemust have α = 0 or/and β = 0 to create a derivative of s in Eq. (12), and there will thusbe two indices 0 contracted with the same antisymmetric tensor ε, either µ = α = 0 or/andν = β = 0. This explains why (12) will not contribute to T 00 in Sec. III below, and whyall derivatives of fn(X) or sn(X) will be canceled in the combination (10). In the staticand spherically symmetric case of Sec. IV, we will see that some s′4 and s′5 do remain, butthey cancel in the particular case X = const. that we will study (and they actually cancelas soon as one assumes X = const., independently of spherical symmetry). It is indeedclear that (12) does not contribute to any derivative of s4 nor s5 if X = const. It is alsoeasy to prove that the only derivatives of functions f or s entering T µν are of the form2(ϕµϕν/M2)f ′(X)L(n,p) when X = const. (or with s′ instead of f ′), because they come fromthe variation of the metric used in the contraction X = −gµνϕµϕν/M2. On the other hand,the scalar current Jµ contains 2(ϕµ/M2)f ′(X)L(n,p), and no other derivative of a functionwhen we assume X = const. Therefore, the linear combination (10) obviously cancels thefew possible f ′ or s′ which occur when X = const.

III. COSMOLOGICAL SELF-TUNING

We consider a homogeneous and isotropic Universe whose metric takes the Friedmann-Lemaıtre-Robertson-Walker (FLRW) form

ds2 = −dt2 + a2(t)

!

dr2

1− kr2+ r2

"

dθ2 + sin2 θ dφ2#

$

, (13)

the parameter k ∈ {−1, 0, 1} determining whether the spatial hypersurfaces are open, flator closed, and we assume consistently that the scalar field ϕ depends only on t. Then itscurrent equation (9b) simply reads ∇0J0 = 0 ⇒ ∂t(a3J0) = 0, and its solution is thusJ0 = C0/a3, where C0 is a constant. This integration constant may be neglected at lateenough times, when the scale factor a becomes very large, and we will thus just solve forJ0 = 0 in the following (keeping in mind that an extra C0/a3 may be added to it).

Once the matter field equations are taken into account, i.e., ∇µJµ = 0 in the presentcase, it is well known that only the time-time-component of the Einstein equations (9a)needs to be solved. Indeed, the covariant conservation of the Einstein tensor ∇µGµν = 0

implies Gij = −gij%

G00 + G00/(3H)&

, where a dot denotes time differentiation andH ≡ a/a,

therefore the spatial components of the Einstein tensor are automatically solved once G00 is,

9

J0 ⇠ 1

a3

FLRW ansatz

Assume and H = 0 k = 0

Algebraic cosmological equations!

Cosmology

It is not difficult to get self-tuning for any parameter M in the action, by adjusting the action

M ⌧ H

M ⇡ H

M � H

Local solutions

Local solutions

Cosmology is fine but what about local solutions?

Local solutions

and no longer for a vacuum energy whose quantum prediction cannot have the observedorder of magnitude.

As underlined above Eqs. (18), L(4,0) and L(4,1) give almost identical field equations.Predicting an observed Λeff independent from Λbare is thus obviously possible too in theL(2,0) + L(4,1) subclass of models with α = γ, i.e., with f2 = k2Xα and s4 = κ4Xα. Theparticular case α = 0, corresponding to f2 = −1 and s4 = 1, was actually the first modelfound with this property, in Ref. [26]. Another example of this kind is given in Eqs. (21)above, where Λeff = M2/4 is indeed independent from Λbare. This behavior can also beobtained in most other combinations of Lagrangians L(n,p), for instance with f2 = k2Xα

and either f3 = k3Xα−1/2, or f5 = k5Xα−3/2, or s5 = κ5Xα−1/2 (all other functions beingassumed to vanish). The only particular cases for which it is not possible to predict Λeff

independent from Λbare are again the L(4,0) + L(4,1) andL(5,0) + L(5,1) combinations, that wealready mentioned in the paragraph below Eqs. (18): One always predicts Λeff ∝ Λbare in

the first case, while Λeff ∝ Λ2/3bare in the second.

IV. SCHWARZSCHILD-DE SITTER SOLUTIONS

A. Self-tuning solutions around a spherical body

We now consider the same models as above, with the same cosmological behavior at largedistances, but we study their predictions in the vicinity of a spherical and static massivebody. Are they consistent with the Schwarzschild metric, which is very well tested at the firstpost-Newtonian order in the solar system? We do know that these models generically exhibita Vainshtein mechanism, which reduces the observable scalar-field effects at small enoughdistances. But in the present self-tuning context, we saw in Sec. III that some quantities(like ϕ) can take extremely large values, therefore the backreaction of the scalar field can apriori fully change the behavior of the metric, and solar-system tests are not guaranteed tobe passed. Actually, one might even fear that none of these models is consistent with localtests, in spite of the Vainshtein mechanism.

A large number of works has been devoted in the literature to the Vainshtein mechanismin Galileon theories. However, most of the studies assumed a time-independent scalar field,see e.g. [37–42] (for a recent review on the Vainshtein mechanism see [25]). The Vainshteinmechanism with a time-dependent scalar has been considered in [24, 43], while Ref. [44]studied it in a subclass of beyond Horndeski theories (with L(5,0) = L(5,1) = Λbare = 0).

Our approach is quite different in the present Section. We shall scan the whole classof generalized Horndeski theories to look for a subclass which (i) is able to screen a hugecosmological constant Λbare, and (ii) reproduces the exact Schwarzschild-de Sitter solutionof GR with a small but non-vanishing observed cosmological constant Λeff.

We choose to work in Schwarzschild coordinates

ds2 = −eν(r)dt2 + eλ(r) + r2!

dθ2 + sin2 θ dφ2"

, (22)

and we consider a scalar field of the form

ϕ = ϕct+ ψ(r), (23)

where ϕc is now assumed to be a constant [contrary to Eqs. (14) and (15) above, whichwere valid for any time dependence]. This ansatz (23) allows us to separate time and radial

15

variables in all field equations because of the shift-symmetry of the theory. Indeed, since anundifferentiated ϕ cannot appear in these field equations, time derivatives are transformedinto the constant ϕc (or 0), and we thus get only ordinary differential equations with respectto the radial coordinate. We also define the dimensionless ratio

q ≡ϕc

M, (24)

and the cosmological value of the standard kinetic term (2) reads thus Xc = q2 = const.In the previous section, we saw that the 00-component of the Einstein equations and the

scalar equation were enough to solve all field equations in FLRW. In the present static andspherically symmetric situation, three equation become necessary and sufficient. Indeed,the covariant conservation of the Einstein tensor ∇µGµν = 0 implies now 2eλGθθ/r3 =∂rGrr + (2/r − λ′ + ν ′/2)Grr + ν ′eλ−νG00/2, where a prime denotes radial differentiation,therefore the angle-angle components Gφφ = sin2 θGθθ are automatically solved once the00 and rr components are, while all off-diagonal components vanish identically. As before,when taking into account the energy-momentum of the scalar field, this remains valid forGµν − T µν/M2

Pl instead of Gµν , up to terms proportional to the scalar equation ∇µJµ = 0,because of Eq. (11) implied by the diffeomorphism invariance of action (3).

We give in Appendix B the two relevant Einstein equations and the scalar current. Ac-tually, we also simplified the rr-Einstein equation by combining it with the scalar current asin Eq. (10). In the present spherically symmetric case with ϕ = const., the scalar equation∇µJµ = 0 simply reads ∂r(r2Jr) = 0. Its solution is thus in general Jr = Cr/r2, where Cr

is an integration constant. However, since we assume that there is no bare matter-scalarcoupling in action (3), the scalar field does not have any source term even within matter,therefore this integration constant must vanish (otherwise the scalar field would be singularat r = 0).

Since Eqs. (B1)–(B3) are quite heavy, we checked them again by two independent meth-ods: First by deriving the full covariant equations and specifying them to metric (22); andsecond the “minisuperspace” technique, in which the form (22) is imposed directly withinthe action, and then one varies it with respect to the three fields ν(r), λ(r) and ϕ′(r) = ψ′(r).

Our aim is now to exhibit a subclass of models which is consistent with an exactSchwarzschild-de Sitter metric. We therefore impose so in Eqs. (B1)–(B3), by enforcingthe metric to take the form (22) with

eν = e−λ = 1−rsr− (Hr)2, (25)

where H is the Hubble rate (assumed to be constant), related to the observed cosmologicalconstant by Λeff = 3H2. These field equations (B1)–(B3) then become long expressionsdepending on the radial coordinate r and radial derivatives of the scalar field (23). However,we noticed that an extra hypothesis simplifies them tremendously. In addition to the aboveassumptions (22), (23) and (25), we will also restrict to the case where X ≡ −(ϕλ)2/M2

remains constant everywhere, even in the vicinity of the massive body. This means that wesimply impose

X = Xc, (26)

where Xc = q2 is the constant cosmological value of X . All functions of X enteringEqs. (B1)–(B3) then obviously become constants, whose precise values are still unknown,but which do not depend any longer on the radial coordinate r. Moreover, since we have

16

and no longer for a vacuum energy whose quantum prediction cannot have the observedorder of magnitude.

As underlined above Eqs. (18), L(4,0) and L(4,1) give almost identical field equations.Predicting an observed Λeff independent from Λbare is thus obviously possible too in theL(2,0) + L(4,1) subclass of models with α = γ, i.e., with f2 = k2Xα and s4 = κ4Xα. Theparticular case α = 0, corresponding to f2 = −1 and s4 = 1, was actually the first modelfound with this property, in Ref. [26]. Another example of this kind is given in Eqs. (21)above, where Λeff = M2/4 is indeed independent from Λbare. This behavior can also beobtained in most other combinations of Lagrangians L(n,p), for instance with f2 = k2Xα

and either f3 = k3Xα−1/2, or f5 = k5Xα−3/2, or s5 = κ5Xα−1/2 (all other functions beingassumed to vanish). The only particular cases for which it is not possible to predict Λeff

independent from Λbare are again the L(4,0) + L(4,1) andL(5,0) + L(5,1) combinations, that wealready mentioned in the paragraph below Eqs. (18): One always predicts Λeff ∝ Λbare in

the first case, while Λeff ∝ Λ2/3bare in the second.

IV. SCHWARZSCHILD-DE SITTER SOLUTIONS

A. Self-tuning solutions around a spherical body

We now consider the same models as above, with the same cosmological behavior at largedistances, but we study their predictions in the vicinity of a spherical and static massivebody. Are they consistent with the Schwarzschild metric, which is very well tested at the firstpost-Newtonian order in the solar system? We do know that these models generically exhibita Vainshtein mechanism, which reduces the observable scalar-field effects at small enoughdistances. But in the present self-tuning context, we saw in Sec. III that some quantities(like ϕ) can take extremely large values, therefore the backreaction of the scalar field can apriori fully change the behavior of the metric, and solar-system tests are not guaranteed tobe passed. Actually, one might even fear that none of these models is consistent with localtests, in spite of the Vainshtein mechanism.

A large number of works has been devoted in the literature to the Vainshtein mechanismin Galileon theories. However, most of the studies assumed a time-independent scalar field,see e.g. [37–42] (for a recent review on the Vainshtein mechanism see [25]). The Vainshteinmechanism with a time-dependent scalar has been considered in [24, 43], while Ref. [44]studied it in a subclass of beyond Horndeski theories (with L(5,0) = L(5,1) = Λbare = 0).

Our approach is quite different in the present Section. We shall scan the whole classof generalized Horndeski theories to look for a subclass which (i) is able to screen a hugecosmological constant Λbare, and (ii) reproduces the exact Schwarzschild-de Sitter solutionof GR with a small but non-vanishing observed cosmological constant Λeff.

We choose to work in Schwarzschild coordinates

ds2 = −eν(r)dt2 + eλ(r) + r2!

dθ2 + sin2 θ dφ2"

, (22)

and we consider a scalar field of the form

ϕ = ϕct+ ψ(r), (23)

where ϕc is now assumed to be a constant [contrary to Eqs. (14) and (15) above, whichwere valid for any time dependence]. This ansatz (23) allows us to separate time and radial

15


q ≡ϕc

M, (24)








eν = e−λ = 1−rsr− (Hr)2, (25)



X = Xc, (26)


16


q ≡ϕc

M, (24)








eν = e−λ = 1−rsr− (Hr)2, (25)



X = Xc, (26)


16

X = e−νq2 − e−λϕ′2/M2 in Schwarzschild coordinates (22), where ϕ′ ≡ ∂rϕ = ψ′ denotesthe radial derivative of the scalar field (23), we may also replace any occurrence of ϕ′ by thesquare root4 of

ϕ′2 = eλ!

e−ν − 1"

M2Xc, (27)

which is a known function of r. Its radial derivative also gives us the exact expression of ϕ′′ ≡∂2rϕ. Therefore, thanks to the greatly simplifying hypothesis (26), the field equations (B1)–(B3) now become mere functions of r alone, involving some unknown constants dependingof the functions fn(X), sn(X) and their derivatives (with respect to X , but also evaluatedat X = Xc). Since these field equations must be satisfied at any spacetime point, it is thenstraightforward to extract from them some necessary conditions on the functions fn(X) andsn(X). For instance, an expansion of Eqs. (B1)–(B3) in powers of (r − r0) around anyradius r0 (even r0 = 0) suffices to prove that some combinations of fn(X), sn(X) and theirderivatives must vanish. After having derived such necessary conditions, one may plug themback into Eqs. (B1)–(B3) to check whether they also suffice. If the field equations do notvanish identically, this means that other conditions still need to be imposed. This procedureallowed us to prove that the following conditions are necessary and sufficient for the ansatz(22), (23), (25) and (26) to be consistent with all field equations (B1)–(B3):

−Xf2 + 6

#

H

M

$2%

X2f4 + 2Xs4&

=M2

Pl

M4

!

Λbare − 3H2"

, (28a)

[Xf2]′ + 6

#

H

M

$2%

X2f4 + 2Xs4&′

= 0, (28b)

Xf5 + 2s5 = 0 and [Xf5 + 2s5]′ = 0, (28c)

%

X3/2f3&′

= 0, (28d)

where as before all functions fn and sn depend on X , and a prime denotes differentiationwith respect to X . This is therefore a particular case of Eqs. (15) that we obtained incosmology, which is not a surprise since the asymptotic behavior of the present solution atlarge radii should match with this cosmological solution. But we find here some restrictionswith respect to (15): The f3 function needs to be very precisely tuned at the cosmologicalvalue X = Xc (in order not to contribute to any background equation at this precise value),while f5 and s5 should be related in a specific way at this value of X (again so that theirsum Xf5 + 2s5 does not contribute to any background equation).

It should be underlined that this set of equations (28) only needs to be satisfied at thecosmological value X = Xc, and notably that (28c) and (28d) should not be imposed for allX . Actually, if Eq. (28d) were satisfied for all X , then the Lagrangian L(3,0), Eq. (4b), wouldbe a total derivative, as underlined in Sec. III above, and it would not contribute to anyobservable. On the other hand, the sum L(5,0) + L(5,1), Eqs. (4d) and (4f), would not be atotal derivative even if the two conditions (28c) were imposed for all X . It just happens thatthis combination does not contribute to the field equations when imposing both sphericalsymmetry and X = const., as in the present section. Note that this combination, satis-fying conditions (28c) for all X , is not the Horndeski one either, which would correspond

4 Obviously, the r.h.s. of Eq. (27) needs to be positive for such an equation to make sense, otherwise this

would correspond to unstable configurations. For example, such a situation takes place in the model

considered in [45] when the bare cosmological constant is absent.

17

We look for Schwarzschild-de Sitter solution

Ansatz:

Important technical assumption:

Local solutions

X = e−νq2 − e−λϕ′2/M2 in Schwarzschild coordinates (22), where ϕ′ ≡ ∂rϕ = ψ′ denotesthe radial derivative of the scalar field (23), we may also replace any occurrence of ϕ′ by thesquare root4 of

ϕ′2 = eλ!

e−ν − 1"

M2Xc, (27)

which is a known function of r. Its radial derivative also gives us the exact expression of ϕ′′ ≡∂2rϕ. Therefore, thanks to the greatly simplifying hypothesis (26), the field equations (B1)–(B3) now become mere functions of r alone, involving some unknown constants dependingof the functions fn(X), sn(X) and their derivatives (with respect to X , but also evaluatedat X = Xc). Since these field equations must be satisfied at any spacetime point, it is thenstraightforward to extract from them some necessary conditions on the functions fn(X) andsn(X). For instance, an expansion of Eqs. (B1)–(B3) in powers of (r − r0) around anyradius r0 (even r0 = 0) suffices to prove that some combinations of fn(X), sn(X) and theirderivatives must vanish. After having derived such necessary conditions, one may plug themback into Eqs. (B1)–(B3) to check whether they also suffice. If the field equations do notvanish identically, this means that other conditions still need to be imposed. This procedureallowed us to prove that the following conditions are necessary and sufficient for the ansatz(22), (23), (25) and (26) to be consistent with all field equations (B1)–(B3):

−Xf2 + 6

#

H

M

$2%

X2f4 + 2Xs4&

=M2

Pl

M4

!

Λbare − 3H2"

, (28a)

[Xf2]′ + 6

#

H

M

$2%

X2f4 + 2Xs4&′

= 0, (28b)

Xf5 + 2s5 = 0 and [Xf5 + 2s5]′ = 0, (28c)

%

X3/2f3&′

= 0, (28d)

where as before all functions fn and sn depend on X , and a prime denotes differentiationwith respect to X . This is therefore a particular case of Eqs. (15) that we obtained incosmology, which is not a surprise since the asymptotic behavior of the present solution atlarge radii should match with this cosmological solution. But we find here some restrictionswith respect to (15): The f3 function needs to be very precisely tuned at the cosmologicalvalue X = Xc (in order not to contribute to any background equation at this precise value),while f5 and s5 should be related in a specific way at this value of X (again so that theirsum Xf5 + 2s5 does not contribute to any background equation).

It should be underlined that this set of equations (28) only needs to be satisfied at thecosmological value X = Xc, and notably that (28c) and (28d) should not be imposed for allX . Actually, if Eq. (28d) were satisfied for all X , then the Lagrangian L(3,0), Eq. (4b), wouldbe a total derivative, as underlined in Sec. III above, and it would not contribute to anyobservable. On the other hand, the sum L(5,0) + L(5,1), Eqs. (4d) and (4f), would not be atotal derivative even if the two conditions (28c) were imposed for all X . It just happens thatthis combination does not contribute to the field equations when imposing both sphericalsymmetry and X = const., as in the present section. Note that this combination, satis-fying conditions (28c) for all X , is not the Horndeski one either, which would correspond

4 Obviously, the r.h.s. of Eq. (27) needs to be positive for such an equation to make sense, otherwise this

would correspond to unstable configurations. For example, such a situation takes place in the model

considered in [45] when the bare cosmological constant is absent.

17

Simple system of equations:

No radial dependence. Again, algebraic equations!

Local solutions

L2, L(4,0), L(4,1) “Three Graces”

Also three “passive” Lagrangians

Other solutions?

No (?).

Violate some assumptions, non-SdS,!

Add some other less restrictive assumptions

X 6= const

However see details in the paper. 1609.09798

Renormalisation of GN

second Einstein equation (38b) also vanish when this condition is assumed. The two Einsteinequations therefore reduce to those of general relativity when condition (42) is imposed, andSchwarzschild solution is recovered at small distances.

Note that Eq. (42) would be a consequence of the two conditions (28c) we found toget our exact solution of Sec. IV, but it does not suffice to imply both of them. In thepresent approximation scheme, we find thus that less constraints are needed to predict aSchwarzschild solution. It is probable that a higher-order analysis, taking into account firstpost-Newtonian terms in the g00 component of the metric [which are of order (rs/r)2], wouldimply a second condition, and that we would then recover the two of Eqs. (28c). But atthe present linear order in rs, the only conclusion we can draw is that the combination ofLagrangians L(5,0) + L(5,1) only needs to satisfy the single condition (42) to be consistentwith a Schwarzschild metric when they dominate locally, whatever the cosmological behavior[which may depend on other Lagrangians L(n,p)] and even if it yields very large factorsmultiplying the f5 and s5 terms in the local equations.

In conclusion, when L(5,0)+L(5,1) dominate the behavior of ϕ in the vicinity of a massivebody, there are two ways to pass solar-system tests. The first one is similar to the caseof L(3,0) above, namely when the cosmological evolution, depending on other LagrangiansL(n,p), is such that the backreaction (41) is small enough with respect to the mass m of thebody (but this needs some well-chosen functions f5 and s5). The second possibility is tochoose a model satisfying condition (42), which is a subset of Eqs. (28c) found for the exactsolutions of Sec. IV. Then the scalar field does not backreact at all on the metric (when f5and s5 locally dominate) whatever the cosmological solution.

B. Renormalization of Newton’s constant

Although the quantity 2Gm = m/(4πM2Pl) entering Eq. (38a) would be called the

Schwarzschild radius of the body in standard general relativity, one should keep in mindthat in the present class of theories, this is not the coefficient entering the possible O(1/r)terms in −g00 and grr. Indeed, the scalar field also contributes crucially to the behavior ofthe metric, and one does not even predict a Newtonian potential ∝ 1/r in most models.Even in the exact solutions of Sec. IV where the metric happens to take the Schwarzschild-de Sitter form, Eqs. (22) and (25), the Schwarzschild radius rs entering its expression doesgenerically differ from 2Gm.

Let us indeed consider the particular case in which only f2, f4 and s4 dominate at smallenough distances, i.e., the Three Graces of Eqs. (28). Let us also assume that X = q2 =const., like in Sec. IV, which implies

ϕ′2 = eλ(e−ν − 1)M2q2 = M2q2rs/r +O(r2s/r2) +O(Λeffr

2). (43)

Note that this means we always have ϕ′2 ≪ ϕ2 = M2q2, i.e., the condition we assumedto make the expansions of Sec. VA, even in the cases where |ϕ| will be predicted to beextremely small with respect to the Planck mass.

Then, the constant contributions to Eq. (38a) (neglecting those ∝ r3 which dominate atlarger distances) imply

rs =2Gm

1 + 4!

MMPl

"2

X1/2 [X5/2f4 + 2X3/2s4]′, (44)

24

where the prime denotes derivation with respect toX . This is equivalent to a renormalizationof Newton’s constant G by the denominator of (44). This renormalization does dependon the cosmological background via X , but note that it is body independent. In otherwords, it cannot be distinguished from general relativity by local experiments, even byequivalence principle tests involving three bodies or more. It suffices that the ratio of thebare gravitational constant G and the denominator of (44) take the experimental value ofNewton’s constant. [Note that we are talking here only of the non-observable effect causedby this renormalization of G. There may exist other deviations from GR in three-bodysystems, for instance preferred-frame effects, that we do not discuss in the present paper.]

In the realistic situation where the observed Λeff is much smaller than Λbare, the added 1in the denominator of Eq. (44) is generically negligible. It is indeed dominated by the secondterm involving functions of X , which is of the same order of magnitude as those enteringthe cosmological equations (15), or more precisely Eqs. (28a) and (28b) in the present ThreeGraces. Combining these equations with (44), we thus generically predict

!

MbarePl

"2Λbare ∼

!

M effPl

"2Λeff, (45)

up to O(1) numerical factors, where MbarePl means our previous notation MPl, while M eff

Pl

is the numerical value corresponding to the actually measured Newton’s constant. For in-stance, in example (19), one gets 5

!

MbarePl

"2Λbare = 3

!

M effPl

"2Λeff, while example (20) gives

3!

MbarePl

"2Λbare =

!

M effPl

"2Λeff. Let us recall that quantum field theory predicts the value of

the vacuum energy density from the matter action Smatter of Eq. (3). Although we decide to

write it as a product!

MbarePl

"2Λbare in this action, it is a priori unrelated to Newton’s con-

stant nor to the observed accelerated expansion of the Universe. The cosmological constantproblem is precisely that the measured values of G (e.g. by Cavendish experiments) and of

the cosmological constant (e.g. from type-Ia supernovae data) give a product!

M effPl

"2Λeff

much too small, by many orders of magnitude, with respect to the predicted vacuum energydensity

!

MbarePl

"2Λbare. In the present scenario, Eq. (45) implies thus that the cosmological

constant problem is not solved at all, and not even alleviated: The observable quantity!

M effPl

"2Λeff actually keeps the same order of magnitude as the huge bare vacuum energy

density!However, the generic behavior (45) is no longer valid if the denominator of Eq. (44) is

not large, and this can happen without any fine tuning if the functions f4 and s4 are chosenso that

#

X5/2f4 + 2X3/2s4$′= 0, (46)

at X = Xc. This condition obviously reduces the space of allowed models, but it does notneed any large nor small dimensionless number to be imposed. The combination X5/2f4 +2X3/2s4 itself must not vanish, otherwise the field equations (28a) and (28b) cannot besatisfied (unless f2 ∝ 1/X , meaning that L(2,0), Eq. (4a), is a second bare cosmologicalconstant). We must thus choose

X5/2f4 + 2X3/2s4 = const. (47)

Many possibilities exist in which f4 and s4 almost compensate each other apart from thisconstant, but they all give the same physics both in the cosmological framework of Sec. IVand in our exact solutions for spherical symmetry of Sec. V. It suffices thus to consider thesimplest cases of f4 = k4X−5/2 and/or s4 = κ4X−3/2, where k4 and κ4 are dimensionless

25

What is the value of the integration constant?

Generically:

So the CC problem is not solved!


But!



!

MbarePl

"2Λbare ∼

!

M effPl

"2Λeff, (45)


Pl


!

MbarePl

"2Λbare = 3

!

M effPl


3!

MbarePl

"2Λbare =

!

M effPl




MbarePl




M effPl

"2Λeff


!

MbarePl



M effPl




#

X5/2f4 + 2X3/2s4$′= 0, (46)


X5/2f4 + 2X3/2s4 = const. (47)


25



!

MbarePl

"2Λbare ∼

!

M effPl

"2Λeff, (45)


Pl


!

MbarePl

"2Λbare = 3

!

M effPl


3!

MbarePl

"2Λbare =

!

M effPl




MbarePl




M effPl

"2Λeff


!

MbarePl



M effPl




#

X5/2f4 + 2X3/2s4$′= 0, (46)


X5/2f4 + 2X3/2s4 = const. (47)


25

constants of order 1. Then, Eq. (44) implies that we have strictly MbarePl = M eff

Pl in thissubclass of the Three Graces. In conclusion, the extra condition (46), added to Eqs. (28),now allows us to predict a small observed Λeff while keeping the Planck mass unrenormalized,so that the observed vacuum energy density

!

MbarePl

"2Λeff may be as small as wished.

Note that the six conditions (28) and (46) only need to be satisfied at one value ofX = Xc. Therefore, there still remain six free functions, which do contribute to the evolu-tion of the Universe before it reaches its equilibrium at X = Xc, as well as to the dynamics ingeneric non-symmetric situations or for perturbations around a spherically symmetric solu-tion. However, the only physically relevant terms of the action, for our exact Schwarzschild-de Sitter background, are just a free f2(X) and f4 = k4X−5/2 and/or s4 = κ4X−3/2. All theother functions, including some non-trivial contributions to f4 and s4 which cancel in thecombination (46), are passive for this solution, i.e., do not enter the result.

An example of a model satisfying all conditions (28) and (46) is given in Eqs. (21) above.Since both f4 = k4X−5/2 and s4 = κ4X−3/2 are allowed, it is also possible to use theHorndeski combination, such that F4 = 0 in Eq. (6d). Then all field equations involve atmost second derivatives, which simplifies their analysis (although the third derivatives ofgeneralized Horndeski models with F4 = 0 do not generate an extra degree of freedom, asrecalled in Sec. II). In the present case, F4 = 0 implies k4 = −6κ4, and this corresponds toG4(−M2X) = −2M2κ4X−1/2 in Eq. (6c). Let us choose k2 = κ4 = −1 to simplify. Then thespecific model f2 = s4 = −X−3/2 and f4 = 6X−5/2 is in the Horndeski class, and does notpredict any renormalization of Newton’s constant. It also predicts that the observed Hubblerate H = M/(2

√6) is fully independent from the bare vacuum energy density M2

PlΛbare

involved in action (3), and therefore does not change even after phase transitions possiblymodifying this vacuum energy. On the other hand, this means that the Hubble scale Hneeds to be introduced by hand in the action via the mass scale M , therefore there stillexist some fine-tuning in such a model, although it concerns the mass scale entering theaction of a scalar field instead of the vacuum energy itself. A better model may be forinstance f2 = −X−5/4, f4 = 6X−5/2 and s4 = −X−3/2, which is still in the Horndeskiclass and does not predict any renormalization of Newton’s constant, but which now needsM = (32M2

PlΛbareH2)1/6. In such a case, the mass scale M introduced in the action is thusintermediate between the huge Planck mass and the tiny Hubble rate.

VI. CONCLUSIONS

In this paper, we studied self-tuning in all shift-symmetric beyond Horndeski theories.Our goal is two-fold. First, we demonstrate that the theory does provide a mechanism toalmost fully screen a very large bare cosmological constant entering the action, leaving asmall effective (observable) one consistent with the present accelerated expansion of theUniverse. Second, we select a subclass of beyond Horndeski theories which not only providesuch a self-tuning of the cosmological constant, but also do not contradict Solar system tests.

Our starting point is the beyond Horndeski action (3) with only two mass scales in theaction, the Planck mass MPl and an extra scale M . The theory contains six arbitraryfunctions, which specify the different possible kinetic terms of the scalar field, see Eqs. (4).We then progressively reduce the space of allowed models by imposing different physicalrequirements.

First we show that self-tuning is possible for a generic combination of beyond HorndeskiLagrangians, provided that the parameterM is adjusted to predict Λeff ≪ Λbare. At this level

26


There is a subclass of Three Graces which does not give renormalisation of Newton’s constant

Example



!

MbarePl





PlΛbare



VI. CONCLUSIONS




26



!

MbarePl





PlΛbare



VI. CONCLUSIONS




26



!

MbarePl





PlΛbare



VI. CONCLUSIONS




26

Black holes

❖ Schwarzschild-de-Sitter solution:!

❖ Note that !

❖ Solution self tunes vacuum cosmological constant. “Action induced" effective cosmological constant appears.!

❖ NB.

Motivation

S =

Zd

4x

p�g

h⇣R� 2⇤� ⌘ (@�)2 + �G

µ⌫@µ�@⌫�

i

<latexit sha1_base64="kp1h/G9+b3zF04hg37qtV95F1OY=">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</latexit><latexit sha1_base64="kp1h/G9+b3zF04hg37qtV95F1OY=">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</latexit>

f = h = 1� µ

r+

⌘

3�r2

<latexit sha1_base64="vDmo0qgx6MQ2xWDRC4X4rqNBgQ0=">AAACGXicbVBNS8MwGE79nPOr6tFLcAiCONop6sHBwIvHCdYN1jrSLN3CkrYkqTBKf4cX/4oXDyoe9eS/Md0q6OYDIU+e53158z5+zKhUlvVlzM0vLC4tl1bKq2vrG5vm1vatjBKBiYMjFom2jyRhNCSOooqRdiwI4j4jLX94mfuteyIkjcIbNYqJx1E/pAHFSGmpa9oBhPUBrEP7CLqBQDh1eZKlIoOHP2+iUJYeu35+i7ta16xYVWsMOEvsglRAgWbX/HB7EU44CRVmSMqObcXKS5FQFDOSld1EkhjhIeqTjqYh4kR66Xi1DO5rpQeDSOgTKjhWf3ekiEs54r6u5EgN5LSXi/95nUQF515KwzhRJMSTQUHCoIpgnhPsUUGwYiNNEBZU/xXiAdKBKJ1mWYdgT688S5xa9bRqXZ9UGhdFGiWwC/bAAbDBGWiAK9AEDsDgATyBF/BqPBrPxpvxPimdM4qeHfAHxuc3I2qfUw==</latexit><latexit sha1_base64="vDmo0qgx6MQ2xWDRC4X4rqNBgQ0=">AAACGXicbVBNS8MwGE79nPOr6tFLcAiCONop6sHBwIvHCdYN1jrSLN3CkrYkqTBKf4cX/4oXDyoe9eS/Md0q6OYDIU+e53158z5+zKhUlvVlzM0vLC4tl1bKq2vrG5vm1vatjBKBiYMjFom2jyRhNCSOooqRdiwI4j4jLX94mfuteyIkjcIbNYqJx1E/pAHFSGmpa9oBhPUBrEP7CLqBQDh1eZKlIoOHP2+iUJYeu35+i7ta16xYVWsMOEvsglRAgWbX/HB7EU44CRVmSMqObcXKS5FQFDOSld1EkhjhIeqTjqYh4kR66Xi1DO5rpQeDSOgTKjhWf3ekiEs54r6u5EgN5LSXi/95nUQF515KwzhRJMSTQUHCoIpgnhPsUUGwYiNNEBZU/xXiAdKBKJ1mWYdgT688S5xa9bRqXZ9UGhdFGiWwC/bAAbDBGWiAK9AEDsDgATyBF/BqPBrPxpvxPimdM4qeHfAHxuc3I2qfUw==</latexit>

⇤e↵ = �⇣⌘/�<latexit sha1_base64="EA9eQr0VGWYYs+YMWqwx1cZ2Elw=">AAACDnicbVC7SgNBFJ2Nrxhfq5Y2g0GxMW5E1EIhYGNhEcGYQDaE2cndZMjsg5m7YlzyBzb+io2Fiq21nX/j5FFo4oW5HM45lzv3eLEUGh3n28rMzM7NL2QXc0vLK6tr9vrGrY4SxaHCIxmpmsc0SBFCBQVKqMUKWOBJqHrdi4FevQOlRRTeYC+GRsDaofAFZ2iopr3rXhlzizVdhHtMwff79Jzuuw+AjLqmHbie6U077xScYdFpUByDPBlXuWl/ua2IJwGEyCXTul50YmykTKHgEvo5N9EQM95lbagbGLIAdCMd3tOnO4ZpUT9S5oVIh+zviZQFWvcCzzgDhh09qQ3I/7R6gv5pIxVhnCCEfLTITyTFiA7CoS2hgKPsGcC4EuavlHeYYhxNhDkTQnHy5GlQOSwcF5zro3zpbJxGlmyRbbJHiuSElMglKZMK4eSRPJNX8mY9WS/Wu/Uxsmas8cwm+VPW5w9r6JvO</latexit><latexit sha1_base64="EA9eQr0VGWYYs+YMWqwx1cZ2Elw=">AAACDnicbVC7SgNBFJ2Nrxhfq5Y2g0GxMW5E1EIhYGNhEcGYQDaE2cndZMjsg5m7YlzyBzb+io2Fiq21nX/j5FFo4oW5HM45lzv3eLEUGh3n28rMzM7NL2QXc0vLK6tr9vrGrY4SxaHCIxmpmsc0SBFCBQVKqMUKWOBJqHrdi4FevQOlRRTeYC+GRsDaofAFZ2iopr3rXhlzizVdhHtMwff79Jzuuw+AjLqmHbie6U077xScYdFpUByDPBlXuWl/ua2IJwGEyCXTul50YmykTKHgEvo5N9EQM95lbagbGLIAdCMd3tOnO4ZpUT9S5oVIh+zviZQFWvcCzzgDhh09qQ3I/7R6gv5pIxVhnCCEfLTITyTFiA7CoS2hgKPsGcC4EuavlHeYYhxNhDkTQnHy5GlQOSwcF5zro3zpbJxGlmyRbbJHiuSElMglKZMK4eSRPJNX8mY9WS/Wu/Uxsmas8cwm+VPW5w9r6JvO</latexit>

(@�)2 = const

�(t, r) = q t± q

h

p1� h

EB& Charmousis’13

Black holes

Generalisation of a subclass of BH solutions in “John” theory.

to s′5(X) = 34f5(X) (unless we are in the limiting case f5 ∝ X−5/2 and s5 ∝ X−3/2). The

conditions (28c) and (28d), at X = Xc, mean thus that f3, f5 and s5 do not contribute toour Schwarzschild-de Sitter background solution, but they do change the behavior of pertur-bations around this background, and they also change the dynamics before the backgroundreaches its equilibrium configuration.

Equation (28a) may also be rewritten as the expression of the effective (observed) cos-mological constant Λeff = 3H2 in terms of X = Xc and the bare cosmological constant:

Λeff =Λbare +

M4

M2Pl

Xf2

1 + 2!

MMPl

"2(X2f4 + 2Xs4)

. (29)

This form underlines that Xf2 acts as an additive constant to Λbare (recall that if f2 ∝ 1/X ,then the Lagrangian L(2,0) would be another trivial bare cosmological constant), whereasX2f4 and Xs4 can be understood as renormalization factors. However, Eq. (29) cannot beinterpreted so directly because these functions are anyway related via Eq. (28b), and thisexplains notably why we found models in which Λeff is fully independent from Λbare, at theend of Sec. III. In the realistic case where 3H2 = Λeff ≪ Λbare, one may of course neglect3H2 in the r.h.s. of Eq. (28a), and the added 1 in the denominator of Eq. (29) may thus besuppressed.

The conclusion of the present subsection is that a subclass of beyond Horndeski theoriesdoes provide both cosmological self-tuning, and a local metric around a spherical bodywhich is indistinguishable from GR plus a small Λeff. This subclass depends on all sixfunctions fn and sn defining beyond Horndeski theories, Eqs. (4), but they should satisfythe five relations (28) at the background value X = Xc. Three of them, f2, f4 and s4,are responsible for the self-tuning, because Eqs. (28a) and (28b) generically fix both thevalues of Xc and 3H2 = Λeff (see Sec. III for a discussion of the non-generic cases in whichone of those is not predicted). We shall call them the “Three Graces”. The three otherfunctions, f3, f5 and s5, play a passive role both for the cosmological background and thelocal spherically symmetric solution, provided they satisfy Eqs. (28c) and (28d) at X = Xc.Note however that the latter three “stealth” Lagrangians do contribute to the dynamics ofperturbations around our exact solutions, and also to the time evolution of the Universebefore the equilibrium value X = Xc is reached.

B. Black hole solutions

An interesting byproduct of the above Sec. IVA is the existence of regular black holesolutions. Indeed, when conditions (28) are imposed at X = Xc, then the field equationsadmit the exact Schwarzschild-de Sitter solution (22) and (25) — with an observed Λeff muchsmaller than Λbare. If no matter source is assumed for r > rs, the metric has thus the sameform as that of a general relativistic black hole.

Let us summarize here in which conditions such black holes exist. We consider the mostgeneral beyond Horndeski Lagrangian

L(2,0) + L(3,0) + L(4,0) + L(4,1) + L(5,0) + L(5,1), (30)

defined by Eqs. (1) and (4), but we require the following three conditions at the cosmologi-

18

cally imposed value X = Xc = q2 of X ≡ −(ϕλ)2/M2:

Xf5 + 2s5 = 0, (31a)

[Xf5 + 2s5]′ = 0, (31b)

!

X3/2f3"′

= 0. (31c)

On the other hand, the three other functions f2, f4 and s4 are free, and they fix the valueof Xc from Eqs. (28a) and (28b). Note that conditions (31) do not need to be satisfied forall values of X , but only at X = Xc. The metric (22) reads then

eν = e−λ = 1−rsr− (Hr)2, (32)

with Λeff = 3H2 given by Eq. (29). The solution for the scalar field is such that

X = q2 = const., (33)

so that using the ansatz (23), one finds explicitly

ϕ = qMt + ψ(r), (34)

with

ψ′ = ±qM

#

rs/r + (Hr)2

1− rs/r − (Hr)2. (35)

Although this last exact expression may be explicitly integrated, let us quote here only thesolution for a negligible value of H :

ϕ = qM

$

t±%

2√rsr + rs ln

&√r −√

rs√r +

√rs

'(

+O(H2)

)

. (36)

The regularity of such black-hole solutions is easy to understand. First of all, since themetric is of the Schwarzschild-de Sitter form, it is clear that the backreaction of the scalarfield on the metric is everywhere finite, including both on the event and the cosmologicalhorizons. In fact, for these solutions, the energy-momentum tensor of the scalar field (8a)takes precisely the form of a vacuum energy,

Tµν

M2Pl

= (Λbare − Λeff) gµν , (37)

as is obvious from our exact Schwarzschild-de Sitter solution for the metric. Concerning theregularity of the scalar field itself, let us underline that the invariants X and JµJµ, involvingthe derivatives of ϕ, are regular everywhere. Indeed X = q2 = const. by construction, whilethe current Jµ actually fully vanishes for the present black hole solutions. The reason isthat by construction5 Jr = 0, while J0 can be checked to be also proportional to Eq. (28b)in the present case, therefore the invariant J2 = JµJµ vanishes everywhere. The regularityof this norm of the current is an additional condition which becomes important notably if

5 For a static and spherically symmetric black hole with the time-dependence ansatz (34), Ref. [46] proved

that Jr = 0 follows from the 0r-Einstein equation.

19


Xf5 + 2s5 = 0, (31a)

[Xf5 + 2s5]′ = 0, (31b)

!

X3/2f3"′

= 0. (31c)


eν = e−λ = 1−rsr− (Hr)2, (32)


X = q2 = const., (33)


ϕ = qMt + ψ(r), (34)

with

ψ′ = ±qM

#

rs/r + (Hr)2

1− rs/r − (Hr)2. (35)


ϕ = qM

$

t±%

2√rsr + rs ln

&√r −√

rs√r +

√rs

'(

+O(H2)

)

. (36)


Tµν

M2Pl





19

Impose:

Black holes

Solution


Xf5 + 2s5 = 0, (31a)

[Xf5 + 2s5]′ = 0, (31b)

!

X3/2f3"′

= 0. (31c)


eν = e−λ = 1−rsr− (Hr)2, (32)


X = q2 = const., (33)


ϕ = qMt + ψ(r), (34)

with

ψ′ = ±qM

#

rs/r + (Hr)2

1− rs/r − (Hr)2. (35)


ϕ = qM

$

t±%

2√rsr + rs ln

&√r −√

rs√r +

√rs

'(

+O(H2)

)

. (36)


Tµν

M2Pl





19


Xf5 + 2s5 = 0, (31a)

[Xf5 + 2s5]′ = 0, (31b)

!

X3/2f3"′

= 0. (31c)


eν = e−λ = 1−rsr− (Hr)2, (32)


X = q2 = const., (33)


ϕ = qMt + ψ(r), (34)

with

ψ′ = ±qM

#

rs/r + (Hr)2

1− rs/r − (Hr)2. (35)


ϕ = qM

$

t±%

2√rsr + rs ln

&√r −√

rs√r +

√rs

'(

+O(H2)

)

. (36)


Tµν

M2Pl





19

to s′5(X) = 34f5(X) (unless we are in the limiting case f5 ∝ X−5/2 and s5 ∝ X−3/2). The

conditions (28c) and (28d), at X = Xc, mean thus that f3, f5 and s5 do not contribute toour Schwarzschild-de Sitter background solution, but they do change the behavior of pertur-bations around this background, and they also change the dynamics before the backgroundreaches its equilibrium configuration.

Equation (28a) may also be rewritten as the expression of the effective (observed) cos-mological constant Λeff = 3H2 in terms of X = Xc and the bare cosmological constant:

Λeff =Λbare +

M4

M2Pl

Xf2

1 + 2!

MMPl

"2(X2f4 + 2Xs4)

. (29)

This form underlines that Xf2 acts as an additive constant to Λbare (recall that if f2 ∝ 1/X ,then the Lagrangian L(2,0) would be another trivial bare cosmological constant), whereasX2f4 and Xs4 can be understood as renormalization factors. However, Eq. (29) cannot beinterpreted so directly because these functions are anyway related via Eq. (28b), and thisexplains notably why we found models in which Λeff is fully independent from Λbare, at theend of Sec. III. In the realistic case where 3H2 = Λeff ≪ Λbare, one may of course neglect3H2 in the r.h.s. of Eq. (28a), and the added 1 in the denominator of Eq. (29) may thus besuppressed.

The conclusion of the present subsection is that a subclass of beyond Horndeski theoriesdoes provide both cosmological self-tuning, and a local metric around a spherical bodywhich is indistinguishable from GR plus a small Λeff. This subclass depends on all sixfunctions fn and sn defining beyond Horndeski theories, Eqs. (4), but they should satisfythe five relations (28) at the background value X = Xc. Three of them, f2, f4 and s4,are responsible for the self-tuning, because Eqs. (28a) and (28b) generically fix both thevalues of Xc and 3H2 = Λeff (see Sec. III for a discussion of the non-generic cases in whichone of those is not predicted). We shall call them the “Three Graces”. The three otherfunctions, f3, f5 and s5, play a passive role both for the cosmological background and thelocal spherically symmetric solution, provided they satisfy Eqs. (28c) and (28d) at X = Xc.Note however that the latter three “stealth” Lagrangians do contribute to the dynamics ofperturbations around our exact solutions, and also to the time evolution of the Universebefore the equilibrium value X = Xc is reached.

B. Black hole solutions

An interesting byproduct of the above Sec. IVA is the existence of regular black holesolutions. Indeed, when conditions (28) are imposed at X = Xc, then the field equationsadmit the exact Schwarzschild-de Sitter solution (22) and (25) — with an observed Λeff muchsmaller than Λbare. If no matter source is assumed for r > rs, the metric has thus the sameform as that of a general relativistic black hole.

Let us summarize here in which conditions such black holes exist. We consider the mostgeneral beyond Horndeski Lagrangian

L(2,0) + L(3,0) + L(4,0) + L(4,1) + L(5,0) + L(5,1), (30)

defined by Eqs. (1) and (4), but we require the following three conditions at the cosmologi-

18

Metric:

Galileon:

Relativistic stars interior

Interior of stars

Vacuum solutions are as in General Relativity (with different cosmological constant). !

What about solutions inside matter?

equations of motion to suppress the scalar field gradient sourced by massive objects. Indeed,expanding the metric sourced by an object of mass M to Newtonian order as

ds2 = (�1 + 2�) dt2 + (1 + 2 ) �ij dxi dxj , (1.1)

one finds a correction to the Newtonian potential

d�

dr=

GNM

r

1 + 2↵2

✓r

rv

◆n�, (1.2)

where the dimensionless constant ↵ parameterises the coupling of the scalar to matter andn > 0 is model dependent. A solar mass object has rv ⇠ O(0.1 kpc) [20] and so the cor-rection to GR is strongly suppressed in the solar system. In the case of Horndeski theories,Vainshtein screening is fully e↵ective [21–23]. For beyond Horndeski theories, this mechanismworks outside extended bodies but breaks down inside matter [24]. The equations governingNewtonian perturbations were found to be of the form [24–27]

d�

dr=

GNM(r)

r2+⌥1GN

4

d2M(r)

dr2(1.3)

d

dr=

GNM(r)

r2� 5⌥2GN

4r

dM(r)

dr, (1.4)

where M(r) ⌘ 4⇡R r0 s2⇢(s)ds, and the parameters ⌥1 and ⌥2 are non vanishing when the

theory contains beyond Horndeski terms in its Lagrangian.This opens up the possibility of testing beyond Horndeski theories using astrophysical

objects such as stars [25, 26, 28–30] and galaxy clusters [27]. Currently, ⌥1 is bounded inthe range �0.22 < ⌥1 < 0.027 where the lower bound comes from the Chandrasekhar massof white dwarf stars [30] and the upper bound comes from consistency of the minimum massfor hydrogen burning with the lowest mass hydrogen burning star [28, 29]. For later purposeswe note that prior to the white dwarf constraint, Ref. [26] was able to place the lower limit⌥1 > �2/3 by requiring a sensible stellar profile (with a mass density that decreases with theradius). The best constraint on ⌥2 = �0.22+1.22

�1.19 comes from the agreement of the lensingand hydrostatic mass of galaxy clusters [27].

Constraining these parameters is important because they are directly related to thecoe�cients introduced in the context of the e↵ective description of dark energy that includesHorndeski and beyond Horndeski theories [31–33], via [26, 27]:

⌥1 =4↵2

H

c2T (1 + ↵B)� ↵H � 1and ⌥2 =

4↵H(↵H � ↵B)

5(c2T (1 + ↵B)� ↵H � 1). (1.5)

The coe�cients ↵T ⌘ c2T � 1, ↵B and ↵H are defined at the level of the cosmological back-ground solution and characterise the behaviour of cosmological perturbations [33]. In partic-ular, when the theory is purely Horndeski ↵H = 0 and we thus have ⌥1 = ⌥2 = 0. Therefore,constraints on ⌥i directly restrict the allowed “beyond Horndeski” deviations from GR.

The constraints mentioned above all rely on non-relativistic systems. The purpose ofthis paper is to investigate the existence and structure of relativistic stars in these theories.There are several motivations for such a study. First, the equations of motion for beyondHorndeski theories are very non-linear and it is important to verify that static sphericallysymmetric solutions for relativistic stars exist. Second, there are technical issues relating to

– 2 –

“Breaking of Vainshtein mechanism” !Newtonian perturbations: Kobayashi,Watanabe&Yamauchi’14; Koyama&Sakstein’15;!

Saito,Yamauchi,Mizuno,Gleyzes&Langlois’15

⌥1 and ⌥2 are nonzero for beyond Horndeski theory

Neutron stars


• We re-investigate white dwarf stars using the full TOV equations and find that post-Newtonian corrections are important for massive stars, so much so that the Chan-drasekhar mass for ⌥1 < 0 is larger than the GR prediction, in contrast to the non-relativistic case. For this reason, the bounds found using white dwarf stars should berevisited [30].

The paper is organised as follows: we first present a model that exhibits Vainshteinbreaking and study its cosmology in FLRW coordinates in section 2, focusing on exact deSitter solutions, which allows us to perform an exact transformation to Schwarzschild-likecoordinates. In section 3 we examine the structure of static spherically symmetric objects.The sub-horizon weak-field limit is reviewed in order to remind the reader of the ambiguitiesassociated with selecting a branch. The values of GN and �PPN (= 1) are derived and arefound to agree with the non-relativistic treatment. Next, we focus on the full relativisticproblem and find an exact solution for the metric exterior to the star. Using this, we showthat �PPN = 1 and that the Vainshtein breaking solution is the one which has the correctasymptotic limit i.e. that space-time is asymptotically de Sitter. Finally, we derive andnumerically solve the TOV system for relativistic stars using polytropic and realistic equationsof state. We discuss our results and conclude in section 4.

2 Model and cosmological de Sitter solution

For simplicity and concreteness, we will study one of the simplest models which exhibitsVainshtein breaking inside matter2, characterised by the action

S =

Zd4x

p�g

M2

pl

✓R

2� ⇤

◆� k2L2 + f4L4,bH

�, (2.1)

with

L2 = �µ�µ ⌘ X (2.2)

L4,bH = �X⇥(⇤�)2 � (�µ⌫)

2⇤+ 2�µ�⌫ [�µ⌫⇤�� µ��

�⌫ ] , (2.3)

where ⇤ is a (positive) cosmological constant and k2 and f4 are constant coe�cients. Here,we have used the shorthand notations, �µ ⌘ rµ� and �µ⌫ ⌘ rµr⌫�. We note thatM2

pl = (8⇡G)�1 where G is not Newton’s constant GN but must be related to it by matchingto the weak field limit. The Lagrangian L4,bH is one of the two beyond Horndeski termsintroduced in [8], which lead to higher order equations of motion but without su↵ering froman Ostrogradsky instability. The theory (2.1) contains two tensor modes and a single scalarmode, as can be deduced from the general Hamiltonian analysis of [11]3. Note that (2.1)corresponds to the model studied by [25] augmented by a cosmological constant.

Matter, characterised by the energy-momentum tensor Tµ⌫ , is assumed to be minimallycoupled to the metric gµ⌫ that appears in the action (2.1). As a consequence, the energy-momentum tensor satisfies the usual conservation equation

rµTµ⌫ = 0 . (2.4)

2This model is free from the conical singularity that can appear in a special subclass of models investigatedin [50, 51]. Note that, in those papers, ↵H is defined as a local function and thus coincides only asymptoticallywith the standard definition of ↵H , which depends only on the homogeneous cosmological solution.

3Another Hamiltonian analysis, but restricted to L4,bH, was presented in [52], with the conclusion that thetotal number of degrees of freedom was strictly less than four.

– 4 –

EB,Koyama,Langlois,Saito,Sakstein’16

Neutron stars

Cosmology - de Sitter solution

The tensor equations of motion, which generalise Einstein’s equations, can be written in theform

M2pl(Gµ⌫ + ⇤gµ⌫) +Hµ⌫ = Tµ⌫ , (2.5)

where Gµ⌫ is the familiar Einstein tensor and Hµ⌫ represents the new terms derived from L2

and L4,bH. Finally, since the scalar sector of the theory is shift-symmetric, the equation ofmotion for the scalar field can be written in the form

rµJµ = 0 . (2.6)

The explicit expressions for Hµ⌫ and Jµ are rather involved and we will not write theirgeneral form here but simply give their relevant components in a static spherical symmetricgeometry (see [24] for the general equation in beyond Horndeski theories).

We now seek vacuum (i.e. no matter energy-momentum tensor in addition to the cos-mological constant) de Sitter cosmological solutions, expressed in FLRW coordinates,

ds2 = � d⌧2 + e2H⌧�dr02 + r02 d⌦2

2

�, (2.7)

with H constant. The scalar equation of motion reduces to

@⌧ (a3J⌧ ) = 0, (2.8)

which is solved by J⌧ = 0 (the general solution J⌧ / a�3 quickly approaches this particularsolution).

Substituting the explicit expression for the current, one gets the equation

J⌧ = �2�⇣k2 + 12f4H

2�2⌘= 0 . (2.9)

Einstein’s equations give the Friedmann constraint, which reads

3M2plH

2 = M2pl⇤+ k2�

2 + 30f4H2�4, (2.10)

where an over-dot denotes a derivative with respect to cosmic time ⌧ .Replacing � by v0 and introducing the dimensionless quantity �2 ⌘ ⇤/(3M2

plH2), one

finds that the two above equations imply

k2 = �2M2

plH2

v20

�1� �2

�, f4 =

M2pl

6v40

�1� �2

�. (2.11)

In what follows, we will always eliminate k2 and f4 in favour of v0, H and �. This will guar-antee that our local solution is related to a well defined cosmological solution asymptotically.

The FLRW slicing of de Sitter spacetime is not well adapted to study static sphericallysymmetric objects such as stars. It is therefore convenient to work in Schwarzschild-likecoordinates using the transformation

⌧ = t+1

2Hln

⇥1�H2r2

⇤and r0 =

e�Ht

p1�H2r2

r . (2.12)

In terms of the new coordinates t and r, the metric (2.7) reads

ds2 = �(1�H2r2) dt2 +dr2

1�H2r2dr2 + r2 d⌦2

2 , (2.13)

while the scalar field cosmological solution becomes

�(r, t) = v0t+v02H

ln�1�H2r2

�, (2.14)

which now depends on both temporal and radial coordinates. One may check that theseexpressions solve the current and tensor equations in this coordinate system.

– 5 –

� = �(⌧) = v0⌧


M2pl(Gµ⌫ + ⇤gµ⌫) +Hµ⌫ = Tµ⌫ , (2.5)



rµJµ = 0 . (2.6)



ds2 = � d⌧2 + e2H⌧�dr02 + r02 d⌦2

2

�, (2.7)


@⌧ (a3J⌧ ) = 0, (2.8)



J⌧ = �2�⇣k2 + 12f4H

2�2⌘= 0 . (2.9)


3M2plH

2 = M2pl⇤+ k2�

2 + 30f4H2�4, (2.10)


plH2), one


k2 = �2M2

plH2

v20

�1� �2

�, f4 =

M2pl

6v40

�1� �2

�. (2.11)



⌧ = t+1

2Hln

⇥1�H2r2

⇤and r0 =

e�Ht

p1�H2r2

r . (2.12)


ds2 = �(1�H2r2) dt2 +dr2

1�H2r2dr2 + r2 d⌦2

2 , (2.13)


�(r, t) = v0t+v02H

ln�1�H2r2

�, (2.14)


– 5 –

Static coordinates:


M2pl(Gµ⌫ + ⇤gµ⌫) +Hµ⌫ = Tµ⌫ , (2.5)



rµJµ = 0 . (2.6)



ds2 = � d⌧2 + e2H⌧�dr02 + r02 d⌦2

2

�, (2.7)


@⌧ (a3J⌧ ) = 0, (2.8)



J⌧ = �2�⇣k2 + 12f4H

2�2⌘= 0 . (2.9)


3M2plH

2 = M2pl⇤+ k2�

2 + 30f4H2�4, (2.10)


plH2), one


k2 = �2M2

plH2

v20

�1� �2

�, f4 =

M2pl

6v40

�1� �2

�. (2.11)



⌧ = t+1

2Hln

⇥1�H2r2

⇤and r0 =

e�Ht

p1�H2r2

r . (2.12)


ds2 = �(1�H2r2) dt2 +dr2

1�H2r2dr2 + r2 d⌦2

2 , (2.13)


�(r, t) = v0t+v02H

ln�1�H2r2

�, (2.14)


– 5 –

Neutron stars

Local solution?

3 Static Spherically Symmetric Objects

We now introduce an astrophysical object, which we model as a spherical symmetric perfectfluid configuration whose energy-momentum tensor is of the form

Tµ⌫ = diag (�", P, P, P ) , (3.1)

where "(r) and P (r) denote the energy density and pressure, respectively. Introducing thissource modifies the spacetime metric, which we now write as

ds2 = �e⌫(r) dt2 + e�(r) dr2 + r2 d⌦22 . (3.2)

The relevant equations of motion are the following: one needs the time and radialcomponents of the tensor equations of motion, which are

" = M2pl

e��

r2

⇣�1 + e� + r�0

⌘�M2

pl⇤� k2

⇣e�⌫v20 + e��02

⌘

+f4

e�⌫�2�

r2v20

�(�10 + 14r�0)�02 � 20r�0�00�+ e�3�

r2�(2� 10r�0)�04 + 16r�03�00�

�,(3.3)

P = M2pl

e��

r2

⇣1� e� + r⌫ 0

⌘+M2

pl⇤� k2

⇣e�⌫v20 + e��02

⌘

+f4

e�⌫�2�

r2v20

�(2 + 14r⌫ 0)�02 + 4r�0�00�� 10

e�3�

r2�04 �1 + r⌫ 0

��, (3.4)

as well as the equation of motion for the scalar field, Eq. (2.6), which reduces to

@r

hr2e(⌫+�)/2Jr

i= 0 , (3.5)

implying that Jr = 0. Substituting the explicit expression of the radial component of thecurrent, we get the equation

Jr =8f4e�3�

r2⇥1 + r⌫ 0

⇤�03 + 2e�2��⌫

k2e

�+⌫ � f4v20

5⌫ 0 + �0

r

��0 = 0 . (3.6)

Note that if Jr = 0 the time-radial component of the metric equations of motion is auto-matically satisfied since it is proportional to Jr for the ansatz assumed in this paper [53].Finally, the energy-momentum tensor rµT

µ⌫ = 0 yields

⌫ 0 =2P 0

"+ P, (3.7)

where a prime denotes a derivative with respect to r.Far from the star the solution must asymptotically approach the cosmological solution

(2.13)-(2.14):

⌫ ⇡ �� ⇠ ln�1�H2r2

�, � ⇠ v0t+

v02H

ln�1�H2r2

�for r⇤ ⌧ r < H�1 . (3.8)

Note that the coordinate r is bounded by the value rH = H�1, corresponding to de Sitterhorizon.

– 6 –

3 Static Spherically Symmetric Objects

We now introduce an astrophysical object, which we model as a spherical symmetric perfectfluid configuration whose energy-momentum tensor is of the form

Tµ⌫ = diag (�", P, P, P ) , (3.1)

where "(r) and P (r) denote the energy density and pressure, respectively. Introducing thissource modifies the spacetime metric, which we now write as

ds2 = �e⌫(r) dt2 + e�(r) dr2 + r2 d⌦22 . (3.2)

The relevant equations of motion are the following: one needs the time and radialcomponents of the tensor equations of motion, which are

" = M2pl

e��

r2

⇣�1 + e� + r�0

⌘�M2

pl⇤� k2

⇣e�⌫v20 + e��02

⌘

+f4

e�⌫�2�

r2v20

�(�10 + 14r�0)�02 � 20r�0�00�+ e�3�

r2�(2� 10r�0)�04 + 16r�03�00�

�,(3.3)

P = M2pl

e��

r2

⇣1� e� + r⌫ 0

⌘+M2

pl⇤� k2

⇣e�⌫v20 + e��02

⌘

+f4

e�⌫�2�

r2v20

�(2 + 14r⌫ 0)�02 + 4r�0�00�� 10

e�3�

r2�04 �1 + r⌫ 0

��, (3.4)

as well as the equation of motion for the scalar field, Eq. (2.6), which reduces to

@r

hr2e(⌫+�)/2Jr

i= 0 , (3.5)

implying that Jr = 0. Substituting the explicit expression of the radial component of thecurrent, we get the equation

Jr =8f4e�3�

r2⇥1 + r⌫ 0

⇤�03 + 2e�2��⌫

k2e

�+⌫ � f4v20

5⌫ 0 + �0

r

��0 = 0 . (3.6)

Note that if Jr = 0 the time-radial component of the metric equations of motion is auto-matically satisfied since it is proportional to Jr for the ansatz assumed in this paper [53].Finally, the energy-momentum tensor rµT

µ⌫ = 0 yields

⌫ 0 =2P 0

"+ P, (3.7)

where a prime denotes a derivative with respect to r.Far from the star the solution must asymptotically approach the cosmological solution

(2.13)-(2.14):

⌫ ⇡ �� ⇠ ln�1�H2r2

�, � ⇠ v0t+

v02H

ln�1�H2r2

�for r⇤ ⌧ r < H�1 . (3.8)

Note that the coordinate r is bounded by the value rH = H�1, corresponding to de Sitterhorizon.

– 6 –

Energy momentum tensor:

Schwarzschild-like ansarz for metric:

Equations of motion…!+ Equation of state

Neutron stars

In the weak filed limit (non-relativistic star):

provided that �2 > 2/5.To select the correct branch with ' ! 0 far from the star, one needs to take into account

the cosmological corrections in eqs. (3.9)-(3.11) in order to determine the asymptotic formof the solutions. Remarkably, as shown in the next section, one can find an exact solutionof eqs. (3.3), (3.4) and (3.6) outside the star. This will enable us to show that the correctsolution is eq. (3.17) with '0 < 0, by matching the exact solution with the sub-horizon weak-field approximation. It is not necessarily the case that an exact solution can be found formore general models and so we provide an alternate method for selecting the correct branchin appendix B for the reader wishing to study such models.

For now, we continue in the weak-field limit and substitute eq. (3.17) into the 00- andrr- equations to find

�⌫ 0(r) =6GM(r)

(5�2 � 2)r2+

(�2 � 1)GM 00(r)

2(5�2 � 2), and (3.18)

��(r) =6GM(r)

(5�2 � 2)r� 5(�2 � 1)GM 0(r)

2(5�2 � 2)r. (3.19)

Our next task is to calculate Newton’s constant GN. Since �⌫/2 coincides with thepotential � in the sub-horizon limit, it is immediate to identify GN by inspection of the firstterm on the right hand side of (3.18). One can also relate �� to the gravitational potential defined in isotropic coordinates (see eq. (1.1)). In appendix A we provide the coordinatetransformation between the two coordinate systems and obtain the relation between �� and . Eventually, the equations (3.18) and (3.19) are found to be equivalent to

d�

dr=

GNM

r2+⌥1GNM

00

4(3.20)

d

dr=

GNM

r2� 5⌥2GNM

0

4r2, (3.21)

with

GN =3G

5�2 � 2(3.22)

⌥1 = ⌥2 ⌘ ⌥ = �1

3

�1� �2

�. (3.23)

Note that since �2 > 2/5 we have �1/5 < ⌥ < 1. In this work we will focus our discussionmainly on the case ⌥ < 0, because it is less constrained than the region ⌥ > 0 [28, 29],but we will show results for both regions for completeness. Outside the object one hasM 00 = M 0 = 0 and so one can see GR is recovered with the Eddington light bending (PPN)parameter �PPN = 1, confirming that the Newtonian limit of this theory agrees with GRoutside extended sources. Equations (3.20) and (3.21) constitute the main results of thissubsection, supplemented by (3.22) and (3.23).

3.2 Exact Vacuum Solution and Cosmological Matching

We now turn our attention to the full relativistic solution. Outside the stellar radius R onehas " = P = 0, in which case the equations eqs. (3.3), (3.4) and (3.6) admit the exactsolution

⌫(r) = ��(r) = ln

✓1� M

r�H2r2

◆and (3.24)

– 8 –

3.1 Sub-Horizon Weak-Field Limit

In order to examine the sub-horizon weak-field limit we expand the metric potentials andscalar as

⌫(r) = ln�1�H2r2

�+ �⌫(r) , (3.9)

�(r) = � ln�1�H2r2

�+ ��(r) and (3.10)

�(r, t) = v0t+v02H

ln�1�H2r2

�+ '(r) , (3.11)

where we expect �⌫ ⇠ �� ⇠ GNM/r ⌧ 1 for an object of mass M . Furthermore, weassume that the cosmological corrections that depend on the small (in the sub-horizon limit)parameter Hr ⌧ 1 are negligible with respect to the perturbations due to the central object,namely we assume H2r2 ⌧ �⌫, �� and v0Hr2/2 ⌧ ' (see appendix B). These assumptionsare valid at su�ciently small radii; using the results obtained below, one can show that thisis true for r ⌧ (GNM/H2)1/3, which is much larger than the stellar radius. Simplificationsfor the scalar field are not so straightforward because non-linearities due to higher derivativesmay be important and so we retain all non-linear terms that are not suppressed by powersof �� or �⌫, and ignore terms such as '04/v40 compared with '02/v20 and r'0'00/v20 since theyare suppressed by extra powers of '0/v0 (see [24] for a discussion on this).

With these simplifications, the 00-component of the tensor equation of motion, Eq. (3.3),becomes

8⇡Gr2" = ��+ r��0 � 5(1� �2)'02

3v20� 10(1� �2)r'0'00

3v20, (3.12)

which can be integrated once to give

�� =2GM

r+

5(1� �2)'02

3v20, (3.13)

where we have introduced the function

M(r) ⌘ 4⇡

Zdr r2" , (3.14)

corresponding to the mass within the radius r.Substituting the above expression for �� into the simplified rr-component, Eq. (3.4),

one gets

�⌫ 0 =2GM

r2+

4(1� �2)'02

3v20r� 2(1� �2)'0'00

3v20, (3.15)

where the pressure P has been neglected because P ⌧ " in the Newtonian limit. This canthen be inserted into the scalar equation Jr = 0, which yields

'02(5�2 � 2)'02

3v20� 4GM

r�GM 0

�= 0 . (3.16)

This equation has three branches of solutions: one with '0 = 0, which gives identical predic-tions to GR, and two characterised by

'02

v20=

3

5�2 � 2

✓2GM

r+

GM 0

2

◆, (3.17)

– 7 –

provided that �2 > 2/5.To select the correct branch with ' ! 0 far from the star, one needs to take into account

the cosmological corrections in eqs. (3.9)-(3.11) in order to determine the asymptotic formof the solutions. Remarkably, as shown in the next section, one can find an exact solutionof eqs. (3.3), (3.4) and (3.6) outside the star. This will enable us to show that the correctsolution is eq. (3.17) with '0 < 0, by matching the exact solution with the sub-horizon weak-field approximation. It is not necessarily the case that an exact solution can be found formore general models and so we provide an alternate method for selecting the correct branchin appendix B for the reader wishing to study such models.

For now, we continue in the weak-field limit and substitute eq. (3.17) into the 00- andrr- equations to find

�⌫ 0(r) =6GM(r)

(5�2 � 2)r2+

(�2 � 1)GM 00(r)

2(5�2 � 2), and (3.18)

��(r) =6GM(r)

(5�2 � 2)r� 5(�2 � 1)GM 0(r)

2(5�2 � 2)r. (3.19)

Our next task is to calculate Newton’s constant GN. Since �⌫/2 coincides with thepotential � in the sub-horizon limit, it is immediate to identify GN by inspection of the firstterm on the right hand side of (3.18). One can also relate �� to the gravitational potential defined in isotropic coordinates (see eq. (1.1)). In appendix A we provide the coordinatetransformation between the two coordinate systems and obtain the relation between �� and . Eventually, the equations (3.18) and (3.19) are found to be equivalent to

d�

dr=

GNM

r2+⌥1GNM

00

4(3.20)

d

dr=

GNM

r2� 5⌥2GNM

0

4r2, (3.21)

with

GN =3G

5�2 � 2(3.22)

⌥1 = ⌥2 ⌘ ⌥ = �1

3

�1� �2

�. (3.23)

Note that since �2 > 2/5 we have �1/5 < ⌥ < 1. In this work we will focus our discussionmainly on the case ⌥ < 0, because it is less constrained than the region ⌥ > 0 [28, 29],but we will show results for both regions for completeness. Outside the object one hasM 00 = M 0 = 0 and so one can see GR is recovered with the Eddington light bending (PPN)parameter �PPN = 1, confirming that the Newtonian limit of this theory agrees with GRoutside extended sources. Equations (3.20) and (3.21) constitute the main results of thissubsection, supplemented by (3.22) and (3.23).

3.2 Exact Vacuum Solution and Cosmological Matching

We now turn our attention to the full relativistic solution. Outside the stellar radius R onehas " = P = 0, in which case the equations eqs. (3.3), (3.4) and (3.6) admit the exactsolution

⌫(r) = ��(r) = ln

✓1� M

r�H2r2

◆and (3.24)

– 8 –

Extra pieces in comparison to GR

Choose the right branch of the solution by matching to !outer (vacuum) solution

Neutron starsRESULTS: negative

Figure 1. The mass-radius relation for the polytropic model (top left), SLy4 EOS (top right panel),BSK20 EOS (bottom left panel) for varying values of ⌥ < 0 indicated in the plots, and the extremecase ⌥ = �0.1 using the SLy4 and BSK20 equations of state (bottom right panel). The region betweenthe gray dashed lines represents the highest mass neutron star observed (M = 2.01± 0.04M�). Notethat the axes on each plot have di↵erent scales.

and ⌥ = �0.05. One can see qualitatively similar features to the polytropic case, namely ahigher maximum mass and a shift to larger radii. Observationally, the most massive neutronstar thus far observed is PSR J0348+0432 with a mass M = 2.01 ± 0.04M� [62], and bothequations of state give stable neutron stars that are consistent with this observation whenthe theory of gravity is GR. One can see that even mild deviations from GR (⌥ = �0.05)predict stars as massive as 3M�. Of course, such predictions are consistent with the highestmass observed neutron star and so one may hope to get constraints by looking at smallermass objects. Indeed, one can see that our model with ⌥ < 0 predicts radii that can be 1km or larger than the GR prediction at fixed mass. Typically, fits to neutron star massesand radii predict radii less than 14 km at 2� for masses between 1 and 2M� [63] and one cansee that the neutron stars predicted by the parameter range studied are consistent with this.Moving to larger values of ⌥, we plot the (not so) extreme value ⌥ = �0.1 in the bottomright panel of fig. 1 where drastic deviations from GR can be seen; the masses can be as largeas 5 M� (or larger) in stark contrast with the current stellar evolution paradigm where the

– 11 –

⌥

Neutron starsRESULTS: positive ⌥

Figure 2. The mass-radius relation for the polytropic model (top left), SLy4 EOS (top right panel),BSK20 EOS (bottom left panel) for varying values of ⌥ > 0 indicated in the plots, and the extremecase ⌥ = �0.1 using the SLy4 and BSK20 equations of state (bottom right panel). The region betweenthe gray dashed lines represents the highest mass neutron star observed (M = 2.01± 0.04M�). Notethat the axes on each plot have di↵erent scales

which allowed us to confirm that the Vainshtein breaking branch is the physical one. Fur-thermore, this shows that the PPN parameter �PPN = 1 so that the theory agrees with GRat the post-Newtonian level.

Turning our attention to the structure of relativistic stars, we derived the Tolman-Oppenheimer-Volko↵ equations and used them to find a relativistic correction to the lowerbound on ⌥1 found by [26] that must be satisfied in order to have static spherically symmetricstellar configurations. In particular, the bound is raised from ⌥1 > �2/3 to ⌥1 > �4/9.We solved the TOV equations for polytropic and two realistic neutron star equations ofstate (SLy4 and BSK20) and consistently found mass-radii relations with larger maximummasses and larger radii at fixed mass than predicted by GR when ⌥1 < 0. Configurationswith ⌥1

>⇠ � 0.05 (note that ⌥1 closer to zero implies smaller deviations from GR) predictmaximum masses M ⇠ 3M� and radii R <⇠ 14 km favoured by observation [63] whereas⌥1

<⇠ � 0.05 predict masses that can be larger than 5M� and radii in excess of 14 km.

– 13 –

Neutron starsMany equations of state

4

The constant v0 = � is the time-derivative of the cos-mological scalar field that satisfies the Friedmann equa-tions. Substituting all this into the equations of motionwith the star described by a perfect fluid, and eliminat-ing the scalar ' in the sub-Horizon limit (see [45] for thedetails), one obtains a system of di↵erential equations for�⌫ and �� (the mTOV system) that can be solved givenan equation of state. These equations, which are givenin Appendix A, are identical to those derived in [45]. Bycomparing these equations with (2), one finds that

⌥1 = ⌥2 ⌘ ⌥ =s� 2

3, (12)

where ⇣ = 6M2plH

2(1� s)/v40 (see Appendix A).At order O("), we obtain an additional equation for

the extra metric function !, of the form

!00 = K1(P, ⇢, ��, �⌫,⌥)!0 +K0(P, ⇢, ��, �⌫,⌥)!, (13)

where K0 and K1 are complicated functions given in Ap-pendix A. Once the mTOV equations for �⌫ and �� havebeen solved, one can use them as inputs to solve thisequation for !. As shown in Appendix A, outside thestar the equation for ! reduces to the one predicted byGR so that

! = ⌦� 2Jr3

(14)

outside the star [56]. Here, J is the angular momentumfrom which one can extract the moment of inertia I =J /⌦. Note that at large distances one has ! ! ⌦ so thatthe space-time is asymptotically de Sitter. Equation (14)implies that

I =R4

2

!0(R)/!(R)

3 +R!0(R)/!(R), (15)

where R is the radius of the star. Equation (13) is ho-mogeneous and so once a solution is found, one is free torescale ! by a constant factor to find another solution.The expression (15) for I is invariant under such a rescal-ing and so we can set the central value !(0) = !c = 1without loss of generality. Furthermore, spherical sym-metry imposes that !0(0) = 0. We solve Equation (13)with these boundary conditions at the centre and find!(R) and !0(R) at the surface of the star. We then com-pute I using Equation (15).

III. COMPACT OBJECTS

In our previous paper [45], we used two equations ofstate to investigate the properties of neutron stars inbeyond Horndeski theories. In this work, we use 32equations of state that have been proposed in the litera-ture; similar investigations have been performed for othermodified gravity theories using some of these equationsof state [58–60]. In computing the I–C relations we donot use equations of state containing quarks or hyperons,nor do we use particularly soft ones such as PAL2.

FIG. 1. The maximum mass for each equation of state forvalues of ⌥ indicated in the figure.

A. Neutron Stars

We begin by examining the properties of Neutron stars.

1. Maximum Mass

Previously [45], we found that the maximum mass forneutron stars can be larger than 2M� and may be largerthan 3M�. The two equations of state we used in [45]can hardly be considered generic and so in figure 1 weplot the maximum mass for each equation of state. Weremind the reader that our equations of state are a faith-ful representation of those used in the literature. Notethat we do not include hyperonic or quark equations ofstate. Clearly, the trend of increasing maximum mass isubiquitous and masses in the range 2M� M 3M�are typical for ⌥ <⇠ � 0.03. For GR, this is more like1.5M� M 2M�. Interestingly, equations of statethat are excluded in GR because they cannot accountfor the observed neutron star of 2M� could be revivedby modifying gravity.

The maximum mass in GR is limited by a causalitycondition i.e. the condition that the sound speed shouldbe dP/ d⇢ 1 (see [61–63]), and so the observation ofneutron stars with masses and radii that violate this maythen point to alternative gravity theories since such ob-jects cannot be accounted for in GR. In figure 2 we plotthe maximum mass and radius for each EOS, and, evi-dently, the majority indeed violate the GR causality con-dition when ⌥ <⇠ � 0.03. It would be interesting to cal-culate the equivalent condition in beyond Horndeski the-ories but this lies well beyond the scope of the present

Maximum mass for each EoS.

Sakstein,EB,Koyama,Langlois,Saito’16

Neutron stars

The maximum mass and radius for each EoS.

Sakstein,EB,Koyama,Langlois,Saito’16 5

FIG. 2. The maximum mass and radius for each equationof state. The values of ⌥ are the same as in figure 1. Thelight gray shaded region shows the condition for causality inGR i.e. the condition for the sound speed to be 1 andassumes that the heaviest observed neutron star has a massof 2.01M

�

. The dark gray region corresponds to objects thatwould be more compact than black holes i.e. R < 2GNM .

work7.

2. I–C Relations

As mentioned in the introduction, [18] have found anapproximately universal relation between the dimension-less moment of inertia I = Ic2/GNM

3 and the compact-ness C = GNM/R of the form

I = a1C�1 + a2C�2 + a3C�3 + a4C�4; (16)

In what follows, we will fit our modified gravity modelsto a relation of this form. In the upper left panel offigure 3 we plot the I–C relation for individual stellarmodels for GR and show the best-fitting relation foundby [18] and our own, whose coe�cients are given in tableI. One can see that our relation agrees well with thatof [18]8. In the upper right panel we show the residuals

7 The calculation in beyond Horndeski theories is not as straight-forward as repeating the GR calculation using the mTOV equa-tions. The scalar degree of freedom and matter are kineticallymixed, and so one must find the speed of scalar and density wavesby diagonalising perturbations about the equilibrium structure(this is similar to what must be done to derive the speed of cos-mological perturbations [25]). Furthermore, the GR condition isderived by conjecturing that the equation of state P = ⇢ � ⇢0,for which dP/d⇢ = 1, produces maximally compact stars. It isnot clear that this remains the case in beyond Horndeski theo-ries since the mTOV equations contain new terms that dependon the derivative of the density. Such terms are absent in GR.Finding the maximally compact EOS would require a detailednumerical study similar to [64, 65].

8 We have not shown their best-fitting coe�cients for clarity rea-sons but if one compares the two one finds small di↵erences. This

�I/I = (Ifit � I)/I; one can see that these are less than10% and that there is no clear correlation with C. Weplot the equivalent figures for ⌥ = �0.03 and ⌥ = �0.05in the middle and lower panels of figure 3 and give thecoe�cients for the fitting functions in table I. Evidently,a similar (approximately) universal relation holds in bothcases.The coe�cients in the table by themselves are not par-

ticularly illuminating and a cursory glance does not re-veal whether the di↵erences between the relations for thedi↵erent theories are significant or not. This is partly be-cause the fitting function typically used is phenomenolog-ical and it is not clear how much degeneracy there is be-tween the free parameters. For this reason, we have plot-ted two figures better suited to show that the di↵erencesbetween the GR and beyond Horndeski theories is signif-icant. In figure 4 we plot all three relations on the sameaxes. Evidently, there is a marked di↵erence between thethree. To quantify this, in figure 5 we plot the quantity�(⌥, C) ⌘ (I⌥(C) � IGR(C))/IGR(C), where IGR is ourbest-fitting I–C relation for GR and I⌥ is the equivalentrelation for a beyond Horndeski theory with parameter⌥. We also plot the quantity �GR ⌘ (IGR � IGR)/IGR

where IGR is the best-fit relation found by [18]. Wealso plot the scatter in the best-fitting relation for allthree theories. The di↵erence between the GR relationsis commensurate with the scatter in the best-fitting rela-tions whereas the di↵erence between the GR and beyondHorndeski relations is far greater than this ( >⇠ 15%).Therefore a precise measurement of this relation has thepower to discriminate between di↵erent theories. Wenote that many alternative theories of gravity, such asmassless scalars coupled to matter and Einstein-Dilaton-Gauss-Bonnet, predict similar relations to GR [59, 66]and can therefore cannot be probed using the I–C rela-tion.

B. Hyperon and Quark Stars

Stars containing particles such as hyperons, kaons, orquarks in a colour-deconfined phase have been posited toexist, and their study is an active and ongoing area of re-search, and several of the equations of state we have usedcontain such particles. In this section we briefly discusshyperonic and quark stars in beyond Horndeski theories,focusing on the hyperon puzzle and the transition fromhyperon to quark stars. The former phenomenon canbe solved by beyond Horndeski theories, whilst the lat-ter remains a feature of theory, just as in GR. We notethat our equations of state containing quarks are of thestrange quark matter (SQM) form and are based on theMIT bag model [67]; they do not contain nucleons. Thus,

is to be expected since we use di↵erent equations of state and adi↵erent code to calculate the stellar models. What is importantis that the two curves match very closely in the region [0.05, 0.40]

Slow rotation Sakstein,EB,Koyama,Langlois,Saito’16


• We re-investigate white dwarf stars using the full TOV equations and find that post-Newtonian corrections are important for massive stars, so much so that the Chan-drasekhar mass for ⌥1 < 0 is larger than the GR prediction, in contrast to the non-relativistic case. For this reason, the bounds found using white dwarf stars should berevisited [30].

The paper is organised as follows: we first present a model that exhibits Vainshteinbreaking and study its cosmology in FLRW coordinates in section 2, focusing on exact deSitter solutions, which allows us to perform an exact transformation to Schwarzschild-likecoordinates. In section 3 we examine the structure of static spherically symmetric objects.The sub-horizon weak-field limit is reviewed in order to remind the reader of the ambiguitiesassociated with selecting a branch. The values of GN and �PPN (= 1) are derived and arefound to agree with the non-relativistic treatment. Next, we focus on the full relativisticproblem and find an exact solution for the metric exterior to the star. Using this, we showthat �PPN = 1 and that the Vainshtein breaking solution is the one which has the correctasymptotic limit i.e. that space-time is asymptotically de Sitter. Finally, we derive andnumerically solve the TOV system for relativistic stars using polytropic and realistic equationsof state. We discuss our results and conclude in section 4.

2 Model and cosmological de Sitter solution

For simplicity and concreteness, we will study one of the simplest models which exhibitsVainshtein breaking inside matter2, characterised by the action

S =

Zd4x

p�g

M2

pl

✓R

2� ⇤

◆� k2L2 + f4L4,bH

�, (2.1)

with

L2 = �µ�µ ⌘ X (2.2)

L4,bH = �X⇥(⇤�)2 � (�µ⌫)

2⇤+ 2�µ�⌫ [�µ⌫⇤�� µ��

�⌫ ] , (2.3)

where ⇤ is a (positive) cosmological constant and k2 and f4 are constant coe�cients. Here,we have used the shorthand notations, �µ ⌘ rµ� and �µ⌫ ⌘ rµr⌫�. We note thatM2

pl = (8⇡G)�1 where G is not Newton’s constant GN but must be related to it by matchingto the weak field limit. The Lagrangian L4,bH is one of the two beyond Horndeski termsintroduced in [8], which lead to higher order equations of motion but without su↵ering froman Ostrogradsky instability. The theory (2.1) contains two tensor modes and a single scalarmode, as can be deduced from the general Hamiltonian analysis of [11]3. Note that (2.1)corresponds to the model studied by [25] augmented by a cosmological constant.

Matter, characterised by the energy-momentum tensor Tµ⌫ , is assumed to be minimallycoupled to the metric gµ⌫ that appears in the action (2.1). As a consequence, the energy-momentum tensor satisfies the usual conservation equation

rµTµ⌫ = 0 . (2.4)

2This model is free from the conical singularity that can appear in a special subclass of models investigatedin [50, 51]. Note that, in those papers, ↵H is defined as a local function and thus coincides only asymptoticallywith the standard definition of ↵H , which depends only on the homogeneous cosmological solution.

3Another Hamiltonian analysis, but restricted to L4,bH, was presented in [52], with the conclusion that thetotal number of degrees of freedom was strictly less than four.

– 4 –

3

II. BEYOND HORNDESKI THEORIES

Beyond Horndeski theories [24, 25] are a very broadclass of scalar-tensor theories that exhibit interestingproperties that make them perfect paragons for alterna-tive gravity theories. The e↵ects of modified gravity arehidden, or screened, in the solar system by the Vainshteinmechanism [48] and so classical tests of gravity based onthe parameterised post-Newtonian (PPN) framework areautomatically satisfied5 but novel deviations from GR,often referred to as Vainshtein breaking, are seen insideastrophysical bodies so that the equations of motion forthe weak-field metric potentials defined by

ds2 = �(1 + 2�) dt2 + (1� 2 ) �ij dxi dxj (1)

are modified to [37–39, 43]

d�

dr=

GNM(r)

r2+⌥1GN

4

d2M(r)

dr2(2)

d

dr=

GNM(r)

r2� 5⌥2GN

4r

dM(r)

dr. (3)

The dimensionless parameters ⌥i characterise deviationsfrom GR of the beyond Horndeski type. They are directlyrelated to the parameters appearing in the e↵ective de-scription of dark energy that controls the linear cosmol-ogy of beyond Horndeski theories [50–52] via [39, 43]:

⌥1 =4↵2

H

c2T (1 + ↵B)� ↵H � 1and

⌥2 =4↵H(↵H � ↵B)

5(c2T (1 + ↵B)� ↵H � 1). (4)

The coe�cients ↵i (see [52] for their definitions) will beconstrained by future cosmological surveys aimed at test-ing the structure of gravity on large scales [53] and soany constraints from small scale probes are complimen-tary to these and may provide orthogonal bounds. When↵H = 0, as is the case in GR but also in Horndeski the-ories, the parameters ⌥i vanish. Currently, ⌥1 is con-strained to lie in the range �0.48 < ⌥1 < 0.027 [40–42, 44] using stellar tests whilst ⌥2 is only weakly con-strained by galaxy cluster tests [43]. This is partly dueto the need for relativistic systems that probe and sothe strong field regime is perfect for placing more strin-gent constraints. The upper bound on ⌥1 is free of thetechnical ambiguities related to the strong field regime(non-relativistic stars are used) and so, for this reason,we focus exclusively on the case ⌥1 < 0 in this work.

5 This is the case for most Vainshtein screened theories althoughit has yet to be shown in full generality. In particular, it has notbeen investigted for theories that exhibit Vainshtein breakingwith the exception of the Eddington light bending parameter �(in all theories [37, 38]) and � (only simple theories such as theone presented in [45] and extended here or those which admitexact Schwarzchild-de Sitter solutions such as theories in thethree graces class [49]). See the discussion in section V.

A. Model

The specific action we will consider is

S =

Zd4x

p�g

M2

pl

✓R

2� k0⇤

◆� L2 + f4L4,bH

�,

(5)with

L2 = k2X +⇣

2X2, (6)

L4,bH = �X⇥(⇤�)2 � (�µ⌫)

2⇤+ 2�µ�⌫ [�µ⌫⇤�� µ��

�⌫ ] ,

(7)

with X=�µ�µ; ⇤ is a (positive) cosmological constant

and k2, ⇣, f4, and k0 are constant coe�cients. The aboveaction belongs to the family of the beyond Horndeski the-ory (“three Graces”), which admits exact Schwarzschild-de-Sitter outside a star, with a time-dependent scalarfield [49]6. Our previous model [45] had k0 = 1 and ⇣ = 0but in order to have a model that does not contain a barecosmological constant we will instead set k0 = 0. Thisaction is by no means the most general beyond Horndeskiaction but contains simple models that exhibit Vainshteinbreaking. The L4,bH term corresponds to a covariantisa-tion of the quartic galileon.The derivation of the mTOV equations, which follows

the procedure set out in [45], is rather long and technical,as is the derivation of the new di↵erential equation wewill solve in this work. For this reason, the main detailscan be found in Appendix A but here we will outline theimportant points briefly. We will follow the method ofHartle and Thorne [56, 57] and write the metric as

ds2 = �e⌫(r) dt2 + e�(r) dr2 + r2�d✓2 + sin2 ✓ d�2

�

� 2"(⌦� !(r))r2 sin2 ✓ dt d�. (8)

This describes the geometry of a space-time containinga slowly rotating star with angular velocity ⌦. The slowrotation is enforced by using a small dimensionless book-keeping parameter " ⌧ 1.At zeroth order, i.e. for " = 0, the metric is static

and spherically symmetric. It is convenient to decom-pose each metric potential, as well as the scalar field,into a cosmological contribution, corresponding to thede Sitter solution (in Schwarzschild coordinates), and acontribution sourced by the star [45]:

⌫ = ln�1�H2r2

�+ �⌫(r) (9)

� = � ln�1�H2r2

�+ ��, (10)

� = v0t+v02H

ln�1�H2r2

�+ '(r). (11)

6 Exact Schwarzschild-de Sitter solutions with a time-dependentscalar field also exist in the Horndeski theory [54, 55].

Slow rotation

Hartle and Thorne method:

3

II. BEYOND HORNDESKI THEORIES

Beyond Horndeski theories [24, 25] are a very broadclass of scalar-tensor theories that exhibit interestingproperties that make them perfect paragons for alterna-tive gravity theories. The e↵ects of modified gravity arehidden, or screened, in the solar system by the Vainshteinmechanism [48] and so classical tests of gravity based onthe parameterised post-Newtonian (PPN) framework areautomatically satisfied5 but novel deviations from GR,often referred to as Vainshtein breaking, are seen insideastrophysical bodies so that the equations of motion forthe weak-field metric potentials defined by

ds2 = �(1 + 2�) dt2 + (1� 2 ) �ij dxi dxj (1)

are modified to [37–39, 43]

d�

dr=

GNM(r)

r2+⌥1GN

4

d2M(r)

dr2(2)

d

dr=

GNM(r)

r2� 5⌥2GN

4r

dM(r)

dr. (3)

The dimensionless parameters ⌥i characterise deviationsfrom GR of the beyond Horndeski type. They are directlyrelated to the parameters appearing in the e↵ective de-scription of dark energy that controls the linear cosmol-ogy of beyond Horndeski theories [50–52] via [39, 43]:

⌥1 =4↵2

H

c2T (1 + ↵B)� ↵H � 1and

⌥2 =4↵H(↵H � ↵B)

5(c2T (1 + ↵B)� ↵H � 1). (4)

The coe�cients ↵i (see [52] for their definitions) will beconstrained by future cosmological surveys aimed at test-ing the structure of gravity on large scales [53] and soany constraints from small scale probes are complimen-tary to these and may provide orthogonal bounds. When↵H = 0, as is the case in GR but also in Horndeski the-ories, the parameters ⌥i vanish. Currently, ⌥1 is con-strained to lie in the range �0.48 < ⌥1 < 0.027 [40–42, 44] using stellar tests whilst ⌥2 is only weakly con-strained by galaxy cluster tests [43]. This is partly dueto the need for relativistic systems that probe and sothe strong field regime is perfect for placing more strin-gent constraints. The upper bound on ⌥1 is free of thetechnical ambiguities related to the strong field regime(non-relativistic stars are used) and so, for this reason,we focus exclusively on the case ⌥1 < 0 in this work.

5 This is the case for most Vainshtein screened theories althoughit has yet to be shown in full generality. In particular, it has notbeen investigted for theories that exhibit Vainshtein breakingwith the exception of the Eddington light bending parameter �(in all theories [37, 38]) and � (only simple theories such as theone presented in [45] and extended here or those which admitexact Schwarzchild-de Sitter solutions such as theories in thethree graces class [49]). See the discussion in section V.

A. Model

The specific action we will consider is

S =

Zd4x

p�g

M2

pl

✓R

2� k0⇤

◆� L2 + f4L4,bH

�,

(5)with

L2 = k2X +⇣

2X2, (6)

L4,bH = �X⇥(⇤�)2 � (�µ⌫)

2⇤+ 2�µ�⌫ [�µ⌫⇤�� µ��

�⌫ ] ,

(7)

with X=�µ�µ; ⇤ is a (positive) cosmological constant

and k2, ⇣, f4, and k0 are constant coe�cients. The aboveaction belongs to the family of the beyond Horndeski the-ory (“three Graces”), which admits exact Schwarzschild-de-Sitter outside a star, with a time-dependent scalarfield [49]6. Our previous model [45] had k0 = 1 and ⇣ = 0but in order to have a model that does not contain a barecosmological constant we will instead set k0 = 0. Thisaction is by no means the most general beyond Horndeskiaction but contains simple models that exhibit Vainshteinbreaking. The L4,bH term corresponds to a covariantisa-tion of the quartic galileon.The derivation of the mTOV equations, which follows

the procedure set out in [45], is rather long and technical,as is the derivation of the new di↵erential equation wewill solve in this work. For this reason, the main detailscan be found in Appendix A but here we will outline theimportant points briefly. We will follow the method ofHartle and Thorne [56, 57] and write the metric as

ds2 = �e⌫(r) dt2 + e�(r) dr2 + r2�d✓2 + sin2 ✓ d�2

�

� 2"(⌦� !(r))r2 sin2 ✓ dt d�. (8)

This describes the geometry of a space-time containinga slowly rotating star with angular velocity ⌦. The slowrotation is enforced by using a small dimensionless book-keeping parameter " ⌧ 1.At zeroth order, i.e. for " = 0, the metric is static

and spherically symmetric. It is convenient to decom-pose each metric potential, as well as the scalar field,into a cosmological contribution, corresponding to thede Sitter solution (in Schwarzschild coordinates), and acontribution sourced by the star [45]:

⌫ = ln�1�H2r2

�+ �⌫(r) (9)

� = � ln�1�H2r2

�+ ��, (10)

� = v0t+v02H

ln�1�H2r2

�+ '(r). (11)

6 Exact Schwarzschild-de Sitter solutions with a time-dependentscalar field also exist in the Horndeski theory [54, 55].

7

Theory a1 a2 a3 a4

GR 0.573221 0.297835 �7.56094⇥ 10�3 7.92444⇥ 10�5

⌥ = �0.03 0.366024 0.277797 �7.62076⇥ 10�3 8.64847⇥ 10�5

⌥ = �0.05 0.327218 0.237928 �5.99316⇥ 10�3 5.89305⇥ 10�5

TABLE I. Coe�cients for the fitting relation (16).

FIG. 4. The I–C relation for GR (black solid curve) andbeyond Horndeski theories with ⌥ = �0.03 (blue, dashed)and ⌥ = �0.05 (red, dotted).

FIG. 5. The fractional di↵erence between the best-ftting I–C relations ((I⌥(C) � IGR(C))/IGR(C)) for ⌥ = �0.03 (blue,solid) and ⌥ = �0.05 (red, dashed). We also show the frac-tional di↵erence between our GR relation and the one foundby [18] (black, dotted) as well as the scatter in all three best-fitting relations (light red dots).

tially softer than pure nucleonic ones (see [71] figure II),which has the consequence that the resulting maximummass of neutron stars is significantly lower than starscontaining nucleons only. In particular, the maximummass predicted by realistic hyperonic equations of state[72–74] lies well below the heaviest presently observedneutron star mass of 2.01± 0.04M� [46]. This apparenttension between nuclear physics and neutron star astron-omy constitutes the so-called hyperon puzzle, and severalpotential resolutions within the realm of nuclear physics

have been proposed.The simplest explanation is our lack of understand-

ing of hypernuclear physics. Hyperon interactions arepoorly understood due in part to calculational di�cul-ties and a lack of experimental data. Hyperon-nucleonand hyperon-hyperon interactions can be repulsive andcan produce the additional pressure needed to supportstars as heavy as 2M� (see [75] for example) althoughthe values of the coupling constants in the theory arenot presently known and are typically chosen in order toachieve the requisite 2M�. Similarly, one can scan theparameter space of e↵ective theories including hyperonsand find parameter choices that give 2M� stars [76]. Ina similar vein, three body interactions (TBIs) are ex-pected to be repulsive and may sti↵en the equation ofstate (EOS) su�ciently to resolve the puzzle (see [68, 71]and references therein). TBIs may also raise the densitythreshold for the appearance of hyperons so that theyare simply not present in the cores of neutron stars [77].Again, there is a lack of experimental data pertaining toTBIs and so whether they can resolve the puzzle remainsto be seen.A more exotic solution is the presence of deconfined

quark matter at high densities (see [78–80] and referencestherein). In this scenario, there are two classes of com-pact stars: quark stars and hyperon stars, the latter be-ing unstable above a threshold mass [81–90]. Above thismass, a first-order phase transition to deconfined quarkmatter [91] occurs in the core and the EOS becomessti↵er, resulting in stars of similar mass but smaller radii.The transition liberates a large amount of energy (1053

erg), which may be the source for gamma ray bursts.The reader is reminded that, in what follows, we modelthis transition using a pure SQM model only so that ourquark star models contain no nucleons.

2. Hyperon and Quark Stars in Beyond Horndeski Theories

In figure 6 we plot the mass-radius relation for anequation of state containing hyperons, GN2NPH, and anequation of state containing quarks only, SQM2, for bothgeneral relativity and beyond Horndeski with ⌥ = �0.05.Focusing on the left panel (GR), one sees that the max-imum mass is around 1.5M� (one finds similar resultsfor more modern equations of state [74]), well below themass of the heaviest compact object presently observed[46]. This is one manifestation of the hyperon problem.One can see that stable quark stars with masses com-patible with this exist in the region where hyperon starsare unstable and so one resolution in GR is that stars

Conclusions

❖ Cosmological screening mechanism!

❖ Schwarzschild-de-Sitter solutions: “Three graces”. Relevant for Solar system tests!

❖ Slightly non-SdS solutions? Maybe (not)!

❖ Renormalisation of Newton’s constant!

❖ Black holes!

❖ Neutron stars can have larger mass

Documents

(Cosmology), black holes and stars in beyond Horndeski theoryjuryokuha/2016_Kyoto_babichev.pdf · (Cosmology), black holes and stars in beyond Horndeski theory Eugeny Babichev LPT