C O S M O L O G Y A N D N E W P H Y S I C S 6 5 3

A more adequate variable which is often used in- stead of time is the redshift,

z

:

z

+ 1 =

a

(

t

0

)

/a

(

t

)

,

(8) where

a

(

t

0

)

is the value of the scale factor at the present time, so the value of the redshift today is

z

= 0

. Fo r adi aba tic expa nsi on,

z

is eq ua l to th e ra ti o of th e cosmi c micro wave backgro und radia tion (CMBR) temperature at some earlier time with respect to its present-day value (see below). Th er e is on e mo re ve ry im po rt an t eq ua ti on , th ou gh not an ind epen den t one , nam ely , the cov ari ant ene rgy

–

momentum conservation,

D

ν

T

ν µ

≡

T

ν µ

;

ν

= 0

.

(9) In our special homogeneous and isotropic case, it looks like

˙

ρ

+ 3

H

(

ρ

+

p

) = 0

(10) and follows from Eqs. (4) and (5). General relativity implies automatic conservation of

T

µν

(9), due to the Einstein equations:

G

µν

≡

R

µν

−

1 2

g

µν

R

= 8

π m

2

Pl

T

µν

.

(11) Indeed, the covariant divergence of the left-hand side identically vanishes and so must

T

ν µ

;

ν

. This is a result of general covariance, i.e., invariance of physics with respect to an arbitrary choice of the coordinate frame. In the recent (and not so recent) literature, there are so me pape rs, where the assum pti on of tim e- dependent

“

constants

”

is con sid ered , in par tic u- lar, time-dependent gravitational coupling constant

G

N

=

G

N

(

t

)

and cosmological constant

Λ(

t

)

(about th e lat te r , se e bel ow Se ct io n 7) . Th e au th or s of these works use there the same standard Einstein equations (11) with nonconstant

m

Pl

. Enforcing the condition of conservation of the Einstein tensor,

G

µν

, and the energy

–

momentum tensor of matter,

T

µν

, the authors derive some relations between

G

N

(

t

)

and

Λ(

t

)

. How ever , thi s pro cedu re is at leas t que sti ona ble. If the Ein ste in equ ati ons are der ive d as usu al by fun c- tional di

ﬀ

erentiation of the total action with respect to metric,

δA/δg

µν

= 0

, then the equation must contain additional terms proportional to (second) derivatives of

G

N

over coordinates. If the least-action principle is rejected, then there is no known way to deduce an expression for the energy

–

momentum tensor of matter. The equations of state are usually parametrized as

p

=

wρ.

(12) In many practically interesting cases, parameter

w

is constant, but it is not necessarily true and it may be a function of time. In this case, to determine

w

(

t

)

, one has to solve dynamical equation(s) of motion for the corresponding

ﬁ

eld(s). Sim ple phys ical sys tem s wit h

w

=

−

1

,

−

2

/

3

,

−

1

/

3

,

0

,

1

/

3

,

1

are known. They are, respectively, vacuum state (vacuum energ y), collect ion of nonin - ter act ing plan e dom ain wal ls, str aigh t cos mic str ing s, nonre lativi stic matt er (with

p



ρ

), rel ati vis tic mat ter (with

p

=

ρ/

3

), and the so-called maximum rigid equation of state (

p

=

ρ

). In the last case, the speed of sound is equal to the speed of light (that is why it is the most rigid). It can be realized by a scalar

ﬁ

eld in the course of cosmological contraction. It is strange that matter with

w

= 2

/

3

is absent in this sequence. One can see from Eq. (4) that, if

w <

−

1

/

3

, the cosmological matter antigravitates despite positive energy densi ty . Corre spon dingly , the Univer se ex- pands with acceleration. It is worth noting, however, th at , if

ρ >

0

, an y ma t t er in a

ﬁ

ni te re gi on of sp ac e has normal attractive gravity. Only in

ﬁ

nitely large pieces of matter may antigravitate. For

“

normal

”

matter,

ρ >

0

and

|

ρ

|

>

|

p

|

, and thus

˙

ρ <

0

, so energy density drops down in the course of expansion, as is naturally expected. However, for the vacuum case, the energy dominance condition,

|

p

|

< <

|

ρ

|

, is not ful

ﬁ

lled

p

vac

=

−

ρ

vac

and the vacuum (or vacuum-like) energy density remains constant despite expansion:

ρ

vac

=

const

.

(13) There might be many more strange states of mat- ter, phantoms, with

w <

−

1

[2, 3]. If such a state wer e reali zed , the ene rgy de nsi ty of thi s kin d of mat ter would rise in the course of expansion. As a result of this rise, gravita tiona l repuls ion would become so strong that everything would turn apart in the future, not only galaxies and stellar bodies, but even atoms and particles. This is the so-called

“

phantom

”

cos- molog y . In all exampl es known to me, a const ant

w <

−

1

appears in some pathological models. However, it is possible that a phantom state could exist only for some

ﬁ

nite time, i.e.,

w

=

w

(

t

)

, and ultimately the system returns to the good old state with

w

≥ −

1

. It is ins tru cti ve to see som e s imp le exa mpl es of the expansion regime (we present them for the spatially

ﬂ

at case of

k

= 0

: (1) Nonrelativistic matter,

p

= 0

:

ρ

∼

1

/a

3

, a

∼

t

2

/

3

, ρ

c

=

m

2

Pl

/

(6

πt

2

)

.

(14) (2) Relativistic matter,

p

=

ρ/

3

:

ρ

∼

1

/a

4

, a

∼

t

1

/

2

, ρ

c

= 3

m

2

Pl

/

(32

πt

2

)

.

(15) (3) V acuum(-like),

p

=

−

ρ

:

ρ

=

const

, a

∼

e

Ht

.

(16)