On the time variation of the cosmological redshift in …research.kek.jp/people/hkodama/LLTB2009/[email protected] distinguish LTB model from ΛCDM model, we consider the time

On the time variation of the cosmological redshift in LTB model

Tomohiro KaiOsaka City University

collaboratorsKenichi Nakao Osaka City University

Chul-Moon Yoo APCTP

Plan of talk

1.Introduction

2.LTB model

3.Time variation of the cosmological redshift of light sources. (a) near the center (b) for all z (c)example of positive

4.Summary and discussion

dz /dt0dz /dt0

dz /dt0

dz /dt 0

1.IntroductionThe observational data of Type Ia supernovae (distance-redshift relation) plays critical roles in modern cosmology.

If we assume the homogeneous and isotropic universe, this observation suggests that the volume expansion of our universe is accelerating. Further, if we assume general relativity, this observation suggests the existence of the dark energy component.

The main current of the cosmological model is the ΛCDM model. ≈0.7

However, no one knows the origin of the dark energy.

M≈0.3

One of other possibilities Inhomogeneous universe

Previous workWe respect the isotropy of the universe, we assume that the observer is located at the symmetry center in LTB dust universe.

We succeeded in constructing the LTB dust universe whose distance-redshift relation is equivalent to that in ΛCDM model. [Prog.Theor.Phys.120 (2008)]

As far as we discuss about the distance-redshift relation, we can't distinguishLTB model from ΛCDM model.

To distinguish LTB model from ΛCDM model, we consider the time variation of the cosmological redshift.

The time variation of the cosmological redshift

Recently, some people are interested in this because of a better understanding of the physical origin of the recent acceleration.

Sandage and McVittie pointed out that one should expect to observea time variation of the cosmological redshift in any expanding spacetime.

K.Lake[astro-ph/0703810]Uzan[Phys.Rev.Lett.100,191303]

In our previous work, by adopting LTB model whose distance-redshift relation is equivalent to that in the ΛCDM model, we calculate the time variation of the cosmological redshift.

dz /dt0

z0

ΛCDM

LTB

2

If we observe whether is positive or negative, we can distinguish the LTB model from the ΛCDM model.

dz /dt0

So, we study the time variation of the cosmological redshiftin LTB model and show some property.

We can obtain this equation by differentiating Einstein equations in LTB model.

R t , r =− M rRt , r 2

z≈ Rr , t

z≈− M r Rr , t 2

0

R r , t :areal radius

However, the sign of the time derivative of the cosmological redshift which we observe is not always negative for all z.

R r , t ≡∂ R r , t ∂ t

z≪1

2.LTB model

LTB solutions are exact solutions to the Einstein equations, which describethe dynamics of a spherically symmetric dust fluid.

line element

ds2=−dt 2∂r Rt , r

2

1−k r r2dr2R2t , r d2

Einstein equations

∂t R2=−k r r2 2M r

R4=

∂rM rR2∂r R

M r , k r :arbitrary functions

Stress energy tensor

T ==0

:4-velocity of a dust particle

Solution

R t , r =6M r1/3t−t B r 2 /3 S x

x=k r r2 t−tB r6M r 2/3

t B r :arbitrary function

S x=cosh−−1

61/3sinh −−−2/3 , x=

−sinh −−−2/3

62/3for x0

S x= 1−cos61/3−sin 2/3

, x=−sin 2/3

62/3for x0

S 0=3/41/3

LTB solutions have three arbitrary functions:M r , k r , t B r

Note that the LTB solution has the gauge freedom of choosing the radial coordinate. M r=M 0 r

3We take

Tanimoto and Nambu's formalism

3.Time variation of the cosmological redshiftWe consider two past-directed outgoing radial null geodesics parameterized by z which is infinitesimally close to each other.

S

O

t1

t 2 t 0

r=rb z t=tb z

r=rb z r zt=t b z t z

We require that the radial coordinate is monotonic with respect to z.

:light source

zz z

z z

zdzdt0

= z

t 0=z− z z

t 0=−

z t 0

t

r

S

t 1

t 2 t 0

r=rb z t=tb z

r=rb z r zt=t b z t z :light source

zz z

z

z

z

dzdt0

= z

t 0= z z −z

t 0=

z t 0

Both case, we can rewrite to be parameterized by z.dz /dt0dzdt0

=− 1 t 0

dzdrb

r z r 0=0

t

r

Past-directed outgoing radial null geodesic equations parametrized by z

drdz

= 1−kr21z ∂t∂r R

dtdz=

−∂r R1z ∂t∂r R

Perturbation equations

d rdz

= −1−kr21z ∂t ∂r R [ ∂r kr

2

21−kr2∂t∂r

2 R∂t ∂r R ] r−1−kr2

1z∂t2∂r R

∂t ∂r R2 t

d tdz

= −11z [ ∂r

2R∂t ∂r R

−∂r R∂t ∂r

2R

∂t∂r R2 ] r− 1

1z [1−∂r R∂t2∂r R

∂t ∂r R2 ] t

dzdt0

near the center

We expand the equation of around z=0.dz /dt0

dzdt0

= dzdt0 z=0 ddz dzdt0 z=0 zO z2

(a)

dzdt0

=−403H0

zO z2

0 : rest mass density at z=0 H 0≡ ∂t RR z=0

By using Einstein equations and geodesic equations, we can reduce to

dzdt0

is negative near the center.

S

O

t 1

t 2 t 0

r=rb z t=tb z

r=rb z r zt=t b z t z :light source

zz z

z z

zrb z , tb z

t

r

rb z r z , tb z t z

for all z(b)dzdt0

d dR

0dzdt0

0

rb z r z rb z r z0

At first we show dz /dt00 r z0

proposition

dzdt 0

=− 1 t 0

dzdrb

r z0

O r

tt 2

t 1 t 0

z z

z

S

rc

rc , tb z t z

rc , tb z z O z

r

rc

z z z

drdz

z , rc , tb z t z

drdz

z z , rc , tb z z

drdz

z , rc , tb z t z drdz

z z , rc , tb z z> for all z

r z 0

We show the sufficient condition for r z0

z

drdz

z , rc , t z z drdz

z z , rc , t z t z >

∂t2∂r R

∂t∂r R2

1z ∂t∂r R t∂r R z z

Requirement

・r is monotonic with respect to z. dr /dz0・R is monotonic with respect to r. ∂r R0

∂t ∂r R0

dz /dt00 r z 0

We expand this inequality and take the first order of it,and rewrite it by Einstein equations and geodesic equations.

∂t2∂r R∂r R

1∂r R

∂t ∂r R2

1z ∂t∂r R t∂r R z z

∂t2∂r R∂r R

1∂r R

∂t ∂r R2

1z ∂t∂r R t∂r R z z

On the other hand, Einstein equations leads to

∂t2∂r R∂r R

=−4R3 ∫0R d dR R3dR∫0R R2dR

d dR

0If ,the left-hand side of the inequality is always negative,

so, above inequality is always satisfied.

The right-hand side above inequality is always positive.

drdz

z , rc , t z z drdz

z z , rc , t z t z >

∂t2∂r R∂r R

1∂r R

∂t ∂r R2

1z ∂t∂r R t∂r R z z

dz /dt00 r z 0

positive

∂t ∂r R0, ∂r R0

∂t2∂r R∂r R

=−4R3 ∫0R d dR R3dR∫0R R2dR0

d dR0

d dR

0dzdt0

0

(c)The case that become positivedzdt0

According to the previous proof, if is negative, there is a possibility that become positive.

d dR

dzdt0

＊Example

k r =0We assume that and

T B 0rr1T B2 r2 [ r

2−r122

r12−r2

2 r12r2

2] r1rr2

T B2 r2

r12r2

2 r2r

t B r =

T B :constant

0.09

0.095

0.1

0.105

0.11

0.115

0.12

0 1 2 3 4 5 6 7 8

-20

-15

-10

-5

0

5

10

0 1 2 3 4 5 6 7 8

dzdt0

z

time derivative of the cosmological redshift rest mass density (t=constant)

r

LTB

EDS

CDM

In this case, become 0 at ,we can't calculate for all z.

∂t ∂r R z≈7.3

Numerical result

4.Summary and discussion

summary・The time variation of the cosmological redshift is negative near the center.

・That the rest mass density monotonically decrease for areal radius is sufficient condition for negative time variation of the cosmological redshift .

・We show an example that the time variation of the cosmological redshift become positive.

discussion・We want to understand the mechanism that the time variation of the cosmological redshift become positive.