Rela/vis/c Stars in dRGT Massive Gravity
Phys. Rev. D93(2016) 124013

Masashi Yamazaki (Nagoya University QG Lab.)In collabora/on with T. Katsuragawa, S. Nojiri, and S. D. Odintsov
The additional constraint rµ I µ r = 0 determines the additional degree of freedom (r).

INTRODUCTION⇣⌘⌘⌫
rµ I µ r = 2 e
2 2e + ⌫ 0 e⌫
1 + 3 ˜1 + ˜2r
✓◆
2 2 +⌫
˜1 2 + ⌫ 0 er erro⇤⇥2 ˜2 r(1 e ) + (1 e + r⌫ 0 )e⌫
0Modified gravity, which is alternative theories of general relativity, is considered for
explaining late-time accelerating universe without cosmological constant.
On the other hand, the short range properties should also be considered for its
consistency. Maximum mass of neutron stars is one of phenomena that is influenced
strongly by the short range properties.
The maximum mass is determined by its gravity
and pressure from inner matters. Therefore
gravity theory and hadron physics model
determine the maximum mass.
From observation, a neutron star whose mass isabout 2M⦿ is discovered. To get such a solution
in modified gravity, we also have to consider
the hydrostatic equilibrium (modified TOV eq.).
If some screening mechanism are active, the result
will be similar to general relativity solutions.
If the mechanisms are not active, the result shouldbe that the mass is larger than 2M⦿ because ofthe consistency. The result show whether the theory
is suitable for astrophysics.

=00p
G 4⇡pr + M0=⌫ = 22GMp+⇢r
1r

m20 = ⇤,

0 = 3,

d4 x

2 22m20det(g) R

2nd order for χ1.
31 =1NUMERICAL ANALYSIS◎ Quark star cases (MIT bag model)
P ( (r)) = c( (r)4B)• c : It depends on chosen mass of strange quark. We used c = 0.28.
• B : bag constant. We use B = 60MeV/fm3 .

41m̃20 r3 I˜r r3˜2 = ˜3 = 0 ) ˜1 =The theory describe a ghost-free massive spin-2 field and the mass can be seen as
cosmological constant. The action is made from the Einstein-Hilbert action plus mass
and interaction terms.

g 1fn enGeneral RelaLvity+ Smatter

n=0Gµ + m20 Iµ =2µI=0µTµ ,Mass［× M⦿］• gµ⌫ , fµ⌫ : dynamical and reference metrics
2•  = 8⇡G : gravitational coupling•n⇣Existence of the solution means existence of usual stars in dRGT massive gravity.
As a first step, we consider the minimal model that the constraint is reduced toa linear equation and can be solved uniquely.

de Rham-Gabadadze-Tolley (dRGT) MASSIVE GRAVITYSdRGT =

4th order for χn , m0 (graviton mass) : free parameters

The maximum mass decrease!

• en (X) : n-th order polynomials of X
The condition for having flat spacetime in entirely vacuum restricts the freeparameters. Moreover redefinition of these are reduced to 3 effective freeparameters.
⇣⌘
21, ˜2 :=, ˜3 = 1 3 ˜1 + ˜2
m̃20 := m20 0 , ˜1 :=
0

m204
Xn en

n=0

⇣pg1f

⌘

= m20 0

Minimal model of dRGT massive gravity

04X

n=0n0en

⇣pg1f

⌘

.

m20 Iµ⌫ ⌘ m̃20 I˜µ⌫ .

Radius［× GM⦿］Our Ansatz and Modiﬁed TOV EquaLons

It is known that the screening mechanism (the Vainshtein mechanism) does notwork in the minimal model. If the mechanism is active, the additional attracting
force is screened and the result becomes similar to general relativity case.

For our purpose, we assume static and spherical dynamical metric and the graviton
mass is taken as cosmological constant m20 = ⇤ . For simplicity, the reference metricis taken as Minkowski metric. We consider perfect fluid as the matter.

We can also see the same result in SLy model (one of the traditional star model).
The minimal model have a tendency that the maximum mass become smaller.

(gµ⌫ dxµ dx⌫ =fµ⌫ dxµ dx⌫ =2⌫(r)

22 (r)

2222edt + edr + r d✓ + sin ✓d 2 ,2220222dt + ( (r)) dr + (r) d✓ + sin ✓d 2 .

T µ ⌫ = ( ⇢(r), p(r), p(r), p(r)),
rµ T µ ⌫ = 0 ) ⌫ 0 (r) =We derive modified TOV equations in dRGT massive gravity. The additional degree of
freedom is determined by additional 4th order algebraic equation. The discriminantshows whether the dRGT massive gravity is suitable for astrophysics.

p0 (r).
p(r) + ⇢(r)

From these assumption, we obtain modified TOV equations. The red letters are
modification from general relativity.
2GM (r)e⌘1,r2m̃
GM 0 = 4⇡G⇢r2 + 0 r2 I˜t t ( ,2⇣
2 (r)

p0 =

4⇡Gpr3 + GM0, ),⌘m̃20 r3 I˜r r ( , ⌫) (p + ⇢),r(r 2GM )
◆✓
◆
✓
◆✓
✓
02GM1 1⌫200
02
1++ ⌫01 p= ⌫ +⌫ +r
r2 r+ m̃2 I˜✓ ✓ ( , 0 , , ⌫).
0RESEARCH POSTER PRESENTATION DESIGN © 2012www.PosterPresentations.com

SUMMARY AND DISCUSSION

2GMr

◆0

The minimal model does not have enough massive neutron stars because of absenceof screening mechanism. We also use the same discussions in the other models if thesolutions exist. We can check whether the screening mechanism works by checking
the maximum mass of neutron stars.
There are future works as following.
• Considering additional algebraic equation’s discriminant and existence of thesolutions in several effective free parameters• Classifying the solution’s branch if the equation can have real solutions• Calculating the maximum mass and comparing with the case of general relativity
Our final purpose is restricting the effective parameter regions that have consistentsolutions as astrophysics phenomenology.