Ghost--Free Massive f(R) Theories Modelled as Effective Einstein Spaces & Cosmic Acceleration

8/13/2019 Ghost--Free Massive f(R) Theories Modelled as Effective Einstein Spaces & Cosmic Acceleration

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f(R)

f(R)

f(R)

f(R)

f(R)

V, dim V =n + m,

n, m 2

N

T V, N : T V = hT V vTV.

{Nai(u)}, N =

Nai(x, y)dxi /ya u= (x, y), u = (xi, ya), h i,j,...= 1, 2,...,n

v

a,b,...= n+ 1, n+ 2,...,n+m,

V = (V,N).


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e= (ei, ea) e

= (ei,ea),

ei = /xi Nai(u)/y

a, ea= a= /ya,

ei =dxi, ea =dya + Nai (u)dx

i.

[e, e] = ee ee = W

e

Wbia = aNbi , W

aji =

aij =ej(N

ai) ei(N

aj),

aij

g

g= g(u)e e =gi(x

k)dxi dxi + ga(xk, yb)ea ea.

(g,N)

: g= 0; T = 0;D : Dg= 0; hT = 0, vT = 0, hvT = 0,

D =hD+vD

D

hv

Z= {T},

T = {

T}.

D = +Z T T = 0, R = {R} R = {R}, D

D

Ric = {R :=R} Ric = {R := R}.

Ric h v R= {Rij :=Rkijk ,Ria:= Rkika,Rai :=Rbaib,Rab :=Rcabc}.

R := gR,

R := g

R =g

ij

Rij+ g

ab

Rab.

V

N

g= {g}

q = {q}

D

R

g,

, MP

(g1q) gq, (

g1q)(

g1q)=

gq, 4

k=0

k ek(g1q) = 3 tr

g1q det

g1q,

k.

ek(X)

X

X= [X] :=tr(X) = X

,

e0(X) = 1, e1(X) =X, 2e2(X) =X2 [X2], 6e3(X) =X

3 3X[X2] + 2[X3],

24e4(X) = X

4

6X

2

[X

2

] + 3[X

2

]

2

+ 8X[X

3

] 6[X

4

]; ek(X) = 0

k >4.

R :=R+ 2 2(3 trg1q detg1q)

S

S=M2P

d4u

|g|[f(R) + mL],

(g,) (g,N,D)

R= R+ Z Ric= Ric +Zic, Z Zic,

D = +Z.

T = 0


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mL(g,N)

mT := 2

|g|

(

|g| mL)

g = mLg + 2

( mL)

g.

S 1f(R) :=df(R)/dR,

R= ,

= m+

f+ ,

m = 1

2M2P

mT, f= (

f

2 1f D2 1f

1f )g+

DD 1f1f

,

= 22[(3 tr

g1q det

g1q)

1

2det

g1q)]g

+2

2{q[(

g1q)1]+ q[(

g1q)1]}.

(g,N,D)

=D Z

(R 12 gR) = 0

T = 0. DT = Q = 0, Q[g,N]

(g,N,D).

f

.

/y3,

y4.

gi = e(xi), ga= ha(x

i, t), N3i =ni(xk), N4i =wi(x

k, t).

y4 =t,

=diag[1= 2, 2= m(xi) + f(xi) + (xi),

3= 4, 4= m(xi, t) + f(xi, t) + (xi, t)]

= (m + f + )g.

=

ee,

(xi)

(xi, t).

f

= diag{(x

i, t)}

mS=d4u

|g| mL


4/8

T =pg+ ( +p)vv

,

p,

v

vv = 1 v = (0, 0, 0, 1)

D

+ = 2 ( m + f + ) = 2 ,

h3= 2h3h4 (m + f + ) = 2h3h4 ,

ni + ni = 0, wi i = 0,

i = h

3i, = h

3

, =

ln |h3|3/2/|h4|

,

= ln |h3/

|h3h4||, := e,

=

1 = /x

1

,

= 2, h

3=4h3.

h

a= 0

= 0 f

wi = (i wi4) ln

|h4|, (i wi4) ln

|h3| = 0, kwi = iwk, ni = 0, ink =kni.

2(...)(xk);h3 h4

t

ni wi.

= m + f +

,

2 =

2|| +

dt 2||

, 2 =

||

||2

dt 2||,

(2)/|| = (2)/.

(iwi4) ln

|h3| = 0,

(i)

=i.

h3[] =

2

4||

h4[] = ()2

2 = |

|2

||dt 2 ||

.

kwi = iwk

wi = i/

=iA,

A(xk, t)

t

nk = 1nk+ 2nk

dy4 h4/(

|h3|)

3,

1nk(xi)

2nk(xi)

2nk = 0

1nk =kn(xi).

W,


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ds2 = e(xk,[ m+ f+ ])[(dx1)2 + (dx2)2] +

2[dy3 + kn dxk]2

4| m + f + |

|[] [ m + f + ]|2

| m + f + |3

2

dt 2 | m + f + |

(dt + iA[] dxi)2.

3

f

= = 0

= 0

f

T = {T[,, ,]} D. kn nk(xi, t)

wi = i/

.

,

.

(xi, t)

(xk, [ m + f + ]) m(xi, t), f(xi, t), (xi, t) m(xi), f(xi), (xi)

kn(xk)

m, f, ,

(xi, t) (t)

(xi, t) (t)

= = = 0,

f(

R)

= 0,

h3 h4

R = g. (

m + f + )g

f

t

t= t(xi,t)

|h4|t/t a(xi,t)

ds2 =a2(xi,t)[i(xk,t)(dxi)2 +h3(xk,t)(e3)2 (e4)2],

i =a2e,a2h3 =h3, e3 =dy 3 +kn dxk,e4 =dt+ |h4|(it+wi).

,

0


6/8

a(xi,t) a(t),h3(xi,t) h3(t)

0 a(xi,t) a(t),

a2h3 = 24|| , 22 = | m+ f+ |+dt 2| m+ f+ |

m+ f+

f(R) =R+ M( T),

T := T+2 2(3 trg1q det

g1q).

1M := dM/d T

H :=

a/

a

a(xi,

t)

a(t)

Nai = {ni, wi(t)}

(t). a(t) a(t)

1 + z =a1(t)

T = T(z),

s(t)

s = (1 +z)Hz.

3H2 +12

[f(z) + M(z)] 2(z) = 0,

3H2 + (1 + z)H(zH) 12

{f(z) + M(z) + 3(1 + z)H2 = 0, (z)z f = 0.

z

1M(z) = 0

z f = 0 M( T)

= 0a

3(1+) =0(1 + z)a

3(1+).

(1 + z)3.

f(R)

:= ln a/a0 = ln(1 +z)

t.

f(R)

(xi, ) = m(xi, ) + f(xi, ) + (xi, )

(xi, )

=/

s =Hs s.

f(R)

f(R) = (H2 +H H)[f(R)] 36H2 4H+ (H)2 +H2H 2f(R)]+2.

() :=H2(),

=(R)

f= 18()[2() + 4()]d2f

dR2 + 6 () +12 () dfdR + 20a3(1+)0 a3(1+)(R).


7/8

t , f(R)

.

R =

. df/dR

f()

= 0.

a() H()

32H2 = 32H20 + 0a3 = 32H20 + 0a30 e3,

H0 0

(xi, )

.

= 12H20 wi, ni 0,

R

() :=H20 + 20a

30 e

3

R = 3() + 12() = 12H20 +

20a30 e

3.

X(1 X)d2f

dX2+ [3 (1+ 2+ 1)X]

df

dX 12f = 0,

1+ 2= 12= 1/6 3= 1/2 3= ln[

210 a30(R 12H

20 )]

X := 3 + R/3H20 .

f=F(X) :=F(1, 2, 3; X), A B

F(X) =AF(1, 2, 3; X) + BX13F(1 3+ 1, 2 3+ 1, 2 3; X).

f

f


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Ghost--Free Massive f(R) Theories Modelled as Effective Einstein Spaces & Cosmic Acceleration

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