Redshift - distance relation in inhomogeneous

Redshift - distance relation in inhomogeneous cosmology G¨ unter Scharf ∗ Physics Institute, University of Z¨ urich, Winterthurerstr. 190 , CH-8057 Z¨ urich, Switzerland

Abstract We continue to study a cosmological model with large-scale inhomogeneity. Working in the cosmic rest frame we determine null geodesics, redshift and area and luminosity distances. Combining the result with Hubble’s law enables us to calculate the distance of the local group of galaxies from the origin r = 0 where the Big Bang has taken place. We obtain a surprisingly small value of about 2 million light years.

Keyword: Cosmology

∗

e-mail: [email protected]

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1

Introduction

In this paper we continue the study of inhomogeneous cosmology in the cosmic rest frame. The perturbative solution of Einstein’s equations investigated in [1] is now crucially tested by determining the corresponding redshift - distance relation. We work in the cosmic rest frame throughout. The big advantage of this frame over comoving coordinates is that it is really a frame, that means a global coordinate system for the whole universe. As a consequence its origin r = 0 is the point where the Big Bang has taken place, and t = 0 is the time 14 billion years ago when this has happened. A natural and meaningful question which now occurs is: Where is r = 0 with respect to our place in the Milky Way ? In [1] this question remained open. Here we will answer it: Surprisingly enough r = 0 is not further away than the Andromeda galaxy ! This follows by taken Hubble’s law into account in the inhomogeneous cosmological model. The paper is organized as follows. In the next section we recall the main results of [1] and use it to determine null geodesics; these are needed to find the light rays in the expanding cosmic gravitational field. In section 3 we consider Maxwell’s equations in the eikonal approximation and calculate the redshift. In section 4 we compute the area and luminosity distances for an observer near the origin r = 0. Combining the result with Hubble’s law enables us to determine the distance R of the observer from the origin. We obtain a surprisingly small value for R of about 2 million light years.

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Null geodesics

We choose spherical coordinates t, r, ϑ, φ in the cosmic rest frame and assume the inhomogeneous metric of the form ds2 = dt2 + 2b(t, r)dt dr − a2 (t, r)[dr2 + r2 (dϑ2 + sin2 ϑdφ2 )].

(2.1)

In [3] we have constructed a perturbative solution of Einstein’s equations given by a(t, r) = αt2/3 + tg1 (r) + h1 (r) + O(α−1 ) (2.2) b(t, r) = αt2/3 Lr + tg2 (r) + h2 (r) + O(α−1 ).

(2.3)

Here α and L are universal constants of integration and the functions g1 , g2 and h1 , h2 are explicitly known. For later use we also list the components of the metric tensor corresponding to the line element (2.1) g00 = 1,

g01 = b(t, r), 2

g11 = −a2 (t, r)

g22 = −r2 a2 (t, r),

g33 = −r2 a2 (t, r) sin2 ϑ,

(2.4)

and zero otherwise. The components of the inverse metric are equal to g 00 =

a2 , D

g 22 = −

g 01 = 1

a2 r 2

,

b , D

g 33 = −

g 11 = −

1 D

1 a2 r2 sin2 ϑ

(2.5) (2.6)

where D = a2 + b2

(2.7)

is the determinant of the 2 × 2 matrix of the t, r components. The nonvanishing Christoffel symbols are given by Γ000 =

bb˙ , D

Γ001 = −

ab a, ˙ D

Γ011 =

1 3 (a a˙ − aba0 + a2 b0 ) D

(2.8)

r sin2 ϑ r (ra3 a˙ + ba2 + rbaa0 ), Γ033 = (ra3 a˙ + ba2 + rbaa0 ) D D b˙ aa˙ 1 Γ100 = − , Γ101 = , Γ111 = (aba˙ + aa0 + bb0 ) D D D r r 1 2 0 1 Γ22 = (raba˙ − a − raa ), Γ33 = (raba˙ − a2 − raa0 ) sin2 ϑ, (2.9) D D 0 a˙ a 1 Γ202 = , Γ212 = + , Γ233 = − sin ϑ cos ϑ (2.10) a a r a˙ a0 1 cos ϑ Γ303 = , Γ313 = + , Γ323 = . (2.11) a a r sin ϑ Here the dot means ∂/∂t and the prime ∂/∂r. The propagation of radiation in the cosmic gravitational field is described by the geodesic equation Γ022 =

dk µ + Γµαβ k α k β = 0 dτ where

(2.12)

dxµ (2.13) dτ is the wave vector and τ is the affine parameter along the ray. We want to determine the null geodesics which starts from a point on the z-axis under kµ =

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a certain angle and lies in a plane φ =const. Then only k 0 , k 1 and k 2 are different from zero and the geodesic equation in leading order read as follows dk 0 L2 r2 a˙ 0 2 Lr 0 1 aa˙ 1 2 + (k ) − 2 ak ˙ k + (k ) + dτ β a β β r + (raa˙ + Lr)(k 2 )2 = 0 β dk 1 Lr a˙ 0 2 a˙ 1 (k ) + 2 k 0 k 1 + Lra˙ + L2 r)(k 1 )2 + − 2 dτ β a βa β r + (Lr2 aa˙ − 1)(k 2 )2 = 0 β

(2.15)

(2.16)

dk 2 a˙ 2 + 2 k0 k2 + k1 k2 = 0 ∂τ a r

(2.17)

β = L2 r2 + 1.

(2.18)

where Equation (2.17) can immediately be integrated k2 =

dϑ C = 2 (ar) dτ

(2.19)

where C is a constant of integration which fixes the angle under which the trajectory starts at the z-axis. In the following we are interested in small C, that means the trajectory is nearly radial. To solve the two remaining equations (2.15-16) we use the ansatz √ K Lr − β 0 1 k = + f0 (t, r), k = K + f1 (t, r). (2.20) a a2 Here K is another integration constant which multiplies the radial null geodesics. The latter follows easily from ds2 = 0,

kµ k µ = 0,

(2.21)

it corresponds to C = 0. The sign of the square root has been chosen in such a way that the light ray moves towards r = 0. For small C we neglect quadratic term in fj , then we obtain from (2.15) the following linear equation df0 KLr a˙ K a˙ C 2 a˙ + 2f0 √ − 2f1 √ =− 3 2 (2.22) 2 dτ a βr β a βa 4

The normalization (2.21) gives another linear equation between f0 and f1 : p

2Kf1 β =

p K C2 − 2f0 (β − Lr β). 2 2 a r a

(2.23)

Substituting this into (2.22) we find the simple equation K a˙ df0 + 2f0 2 = 0 dτ a

(2.24)

with the solution

K1 a2 where K1 is another constant of integration. Now by (2.23) we get the second function f1 f0 =

f1 =

C2 K1 p √ − ( β − Lr). a3 2Ka2 r2 β

(2.25)

(2.26)

Since

dt K = + f0 (2.27) dτ a the constant K1 in (2.25) only enters in the dependence t(τ ). Therefore, without loss of generality we can put K1 = 0 by a redefinition of the affine parameter. Finally we have to check that the second geodesic equation (2.16) is satisfied by (2.25-26). This is indeed true in leading order O(a−3 ), higher orders do not interest us because we have neglected them in the Christoffel symbols already. Summing up the wave vector has the following components k0 =

k0 = k1 = K

K dt = a dτ

√ Lr − β dr C2 √ = + a2 dτ 2Ka2 r2 β k2 =

(2.28) (2.29)

dϑ C = . a2 r2 dτ

We put r0 =

5

C K

(2.30)

and assume this to be positive; its physical meaning will soon become clear. By dividing (2.31) and (2.28) we get dϑ r0 = 2. dt ar

(2.31)

Since this is positive the polar angle ϑ(t) is monotonously increasing with time. We consider a trajectory starting on the z-axis at a point r1 at time t1 so that ϑ(t1 ) = 0. This is the place of our radiating galaxy. To find the light path we divide (2.29) by (2.28) p dr 1 r02  k1 √ = = Lr − β + k0 dt a 2r2 β

(2.32)

and separate the variables: √ dt r2 L2 r2 + 1 √ = dr. a(t) Lr3 L2 r2 + 1 − r2 (L2 r2 + 1) + r02 /2

(2.33)

. We consider the light rays towards the origin r = 0, that means dr < 0. dt Setting this equal to 0 we find from (2.32) the minimal distance of the path from the origin s 2 rmin =

r02 + 2

r04 r2 + 02 , 4 L

(2.34)

so r0 determines this minimal distance. To find the path we must integrate (2.33). To make this integration simple we assume L2 r2  1 (2.35) and expand 1 . 2Lr The usefulness of this approximation was discussed in [1]. We integrate (2.33) from an emission time t1 to the present time T : p

ZT t1

β = Lr +

1 dt 1/3 = (T 1/3 − t1 ) = −2L 2/3 3α αt 6

ZR r1

r3 dr r2 − r02

r 2 − r 2  1 0 2 . 2

= L(r12 − R2 + r02 log

R − r0

(2.36)

Here r1 and R are the radial coordinates of the emitter and observer, respectively. Solving for t1 we obtain the equation t = t(r) of the light path r 2 − r 2 i3 0 . 2 2

h

t = T 1/3 − 3αL(r2 − R2 + r02 log

3

R − r0

(2.37)

The redshift

The propagation of electromagnetic radiation in the expanding universe is described by Maxwell’s equations in a gravitational field. The homogeneous equations are ∇λ Fµν + ∇µ Fνλ + ∇ν Fλµ = 0 (3.1) where ∇ stands for the covariant derivative with respect to our inhomogeneous metric. In addition the field tensor F µν satisfies the inhomogeneous equations ∇µ F µν = 0 (3.2) away from the radiating galaxy. As in Minkowski vacuum, equation (3.1) implies the existence of a vector potential Fµν = ∇µ Aν − ∇ν Aµ

(3.3)

and similarly for the upper indices. Here we can impose the “Lorentz” gauge condition ∇µ Aµ = 0. (3.4) As in classical electrodynamics we write (3.2) in terms of the vector potential ∇µ ∇µ Aν − ∇µ ∇ν Aµ = 0. (3.5) Now we want to commute the derivatives in the second term and use (3.4). Since the covariant derivatives do not commute we obtain an additional term with the Ricci tensor ∇µ ∇µ Aν − Rαν Aα = 0. In cosmology it is sufficient to solve this equation in the eikonal approximation by considering a solution of the form Aν (x) = f ν (x) sin[ϕ(x)]. 7

(3.6)

Here f ν is a slowly varying amplitude, ϕ is the phase which varies from 0 t0 2π over the wave length of the radiation. Since the latter is extremely small compared to cosmic distances, ϕ is big so that we shall consider only the first two orders of ϕ in (3.6). The leading order O(ϕ2 ) gives ∇µ ϕ∇µ ϕ = 0

(3.7)

2∇µ f ν ∇µ ϕ + f ν ∇µ ∇µ ϕ = 0.

(3.8)

and in order O(ϕ) we obtain

As in optics one introduces the wave vector k µ = ∇µ ϕ.

(3.9)

Then the two equations to be solved read k µ kµ = 0

(3.10)

1 k µ ∇µ f ν = − f ν ∇µ k µ . 2 From equation (3.10) we have k µ ∇ν kµ = 0,

(3.11)

(3.12)

and since (3.9) implies ∇ν kµ = ∇µ kν we arrive at k µ ∇µ k ν = k µ Let now xµ (τ ) be the curve

then we have

 ∂k ν

∂xµ



+ Γνµλ k λ = 0.

dxµ = kµ , dτ

(3.13)

(3.14)

d2 xν dτ d2 xν 1 ∂k ν = = ∂xµ dτ 2 dxµ dτ 2 k µ

and (3.13) becomes µ λ d2 xν ν dx dx + Γ = 0. (3.15) µλ dτ 2 dτ dτ This is the geodesic equation (2.12) for xν (τ ) which we have solved in the last section.

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Following G.F.R. Ellis [2] (see also [3]) the redshift is given by νem (uµ k µ )em =1+z = νobs (uµ k µ )obs

(3.16)

where uµ is the 4-velocity of the emitting galaxy and the observer, respectively. The radial velocity of the galaxy we know from [1] equations (6.8)(6.9): p L2 rt u0 = β, u1 = √ . (3.17) β Then, since

√ h √ L2 rt Lr − β i β ω ≡ uµ k = K 1+ a(t) β a(t) µ

(3.18)

the redshift becomes √ p L2 rt  β a(T ) h r02 i 1+z = √ 1+ Lr − β + 2 √ × βa(t) βR a(t) 2r β h

× 1+

i−1 p L2 RT  r02 √ , LR − βR + βR a(T ) 2R2 βR

(3.19)

where T is the present age of the Universe and R > 0 the radial coordinate of the observer. The emission time t follows from the coordinate distance r = r1 of the galaxy by means of (2.37). For later use we introduce the expression p L2 rt  r02  Kr = 1 + Lr − β + 2 √ . (3.20) β 2r β Then the redshift is equal to √ β a(T ) Kr 1+z = √ βR a(t) KR

(3.21)

β R = L2 R 2 + 1

(3.22)

where according to (2.18).

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4

Area and luminosity distances

In this section we must specify the radial coordinate R of the observer. We choose R = rmin (4.1) equal to the minimal distance (2.34) of the light path from the origin r = 0 where the Big Bang has taken place. This greatly simplifies the following calculations because we have dr =0 dt R

(4.2)

KR = 1

(4.3)

so that the redshift (3.21) becomes √ β a(T ) 1+z = √ Kr . βR a(t)

(4.4)

Theoretically the simplest radial distance is the area or angular-diameter distance [4] DA = a(t)r(t) (4.5) where r(t) is the radial coordinate of the galaxy at the time of emission t which can be found by inverting (2.37). Assuming reciprocity [3] DA is related to the luminosity distance DL which is measured by the astronomers by DL = (1 + z)2 DA = a(t)r(t)(1 + z)2 . (4.6) Considering t as a parameter the equations (4.4) and (4.6) determine the distance redshift relation DL = DL (z) in parametric form. We want to calculate the derivative dDL c = dz z=0 H0

(4.7)

which gives the Hubble constant H0 at present time T (we have put the velocity of light c equal to 1). From (4.6) we obtain dt dDL = a(T ˙ ) R + 2a(T )R dz z=0 dz T

(4.8)

where (4.2) has been taken into account. Again using (4.2) we get from (4.4) dz ∂z a˙ = − +∂t Kr |T . = dt T ∂t a T 10

(4.9)

Since ∂t Kr |T = 0 because r(T ) = R is the minimal distance (4.1) we get in leading order O(α) (2.2) dz 2 . (4.10) =− T dt 3T Using 2a a(T ˙ )= (4.11) 3T we finally find dDL = a(T )R. (4.12) dz z=0 Now it is time to put some simple numbers in. Let %crit = 1.878 × 10−29 h2 g/cm3

(4.13)

be the critical density [4] and we assume a Hubble constant h = 0.7,

H0 = 70 km s−1 Mpc−1 .

(4.14)

We use the empirical fact that the corresponding Hubble time 1 = 14 × 109 years = T H0

(4.15)

coincides with the age T of the universe. A realistic density of normal matter at present time T is 1 6πGβR T 2

%m = 0.01 × %crit =

(4.16)

where the last equality is the lowest order result [1] eq.(6.7). This allows to determine βR = L2 R2 + 1 = 44.3 (4.17) which gives LR = 6.6.

(4.18)

Next we assume a radial velocity of our Galaxy of 300 km/sec that means vm = 0.001 =

11

LR a(T )

(4.19)

where the last equality comes from [1] eq.(7.6). Using (4.18) this allows to determine a(T ) = 6.6 × 103 . (4.20) Now we are able to calculate R from (4.12) using (4.7) (4.12) and the Hubble time (4.15) R=

T = 2.1 × 106 ly = 640 kpc. a(T )

(4.21)

This is the distance between our Galaxy and the origin r = 0 where the Big Bang has taken place. It is even smaller than the distance of the Andromeda galaxy which is 2.5 × 106 light years away. From the dipole anisotropy of the CMB one has derived a net velocity of the local group of galaxies of 627 ± 22 km/sec in a direction between the Hydra and Centaurus clusters of galaxies [4] [6]. We expect that the origin r = 0 lies in the opposite direction. The velocity about 600 km/sec seems to be rather high. But this velocity is the sum of the cosmic radial velocity and a local velocity due to the attraction of the Hydra and Centaurus clusters. Since the latter has not been estimated we have made the simple assumption that 300 km/sec is the cosmic velocity (4.19). If we calculate the second derivative d2 DL /dz 2 in the same way we get zero. This shows that a lowest order calculation is not good enough to determine the acceleration parameter q0 .

References [1] Scharf G., 2013, arXiv 1312.2695 [2] Ellis G.F.R., 2009, Gen.Rel.Grav. 41, 581 [3] Ellis G.F.R., Maartens R., MacCallum M.A.H., 2012, Relativistic Cosmology, Cambridge Universety Press [4] Weinberg S. 2008, Cosmology, Oxford University Press [5] Weinberg S. 1972, Gravitation and Cosmology, John Wiley & Sons [6] Kogut A., Lineweaver C., Smoot G.F., Bennett C.L., Banday A. Boggess N.W. et al., 1993, Astrophys.J. 419, 1

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