fulltext

A time-dependent “cosmological constant” phenomenology (*)

S. CAPOZZIELLO(**), R.DERITIS(***) and A. A. MARINO(* **)

Dipartimento di Scienze Fisiche, Università di Napoli, INFN, Sezione di Napoli Mostra d’Oltremare, Pad. 19, I-80125 Napoli, Italy

(ricevuto il 13 Novembre 1996; approvato il 25 Marzo 1997)

Summary. — We construct a cosmological toy model in which a “cosmological constant” depending on time as a step-function is taken into consideration besides ordinary matter. We assume that L takes two values depending on the epoch, and matter goes from a radiation-dominated era to a dust-dominated era. The model is exactly solvable and it can be compared with recent observations.

PACS 98.80.Cq – Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.).

PACS 98.70.Ve – Background radiations.

1. – Introduction

A cosmological-constant term in the energy density of the universe turns out to be taken in serious consideration in today research. Many experimental data on the structure of the present universe are in agreement with models in which the cosmological-constant term contribution in the density parameter V (given by the ratio of the mean mass density to the critical density of the corresponding Einstein-de Sitter model rc4 3 H2O8 pG) is relevant with respect to the matter term. This happens in the

number count of galaxies as well as in the spectrum of the Cosmic Microwave Background Radiation (CMBR) [1-3]. On the one hand, there is the well-known problem concerning the theoretical and experimental limits on the values that the cosmological costant should assume, connected, respectively, with the vacuum expectation value of the energy density and the present observational limits, in the sense that they differ by 120 orders of magnitude [4, 5]. On the other hand, despite these discrepancies, the presence of a cosmological-constant term is requested to avoid

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail: capozzielloHaxpna1.na.infn.it

(***) E-mail: deritisHaxpna1.na.infn.it (*

**) E-mail: marinoHaxpna1.na.infn.it

the problems which are found when comparing the age of the universe as coming from estimates of globular clusters age with the value obtained from a standard model with a unitary density parameter, if one wants to give to the Hubble parameter H0a value in

agreement with the most recent results [2, 6, 7]. As is well known, the value for the age comes out to be too low, while with a cosmological-constant term one is able to find the agreement with the observations. Finally, from the statistic of the gravitational lensing, an upper limit on L comes out, that is its contribution to V seems to be less than 95% [8].

Furthermore, models in which a cosmological constant is present give rise to the inflationary epoch which solves the problems of the cosmological standard model such as the horizon, flatness, entropy problems [9, 10]. In such a context, the present value of the density parameter turns out to be very close to unity. If we consider that the observational estimates of V from the smaller scales of galaxies to scales of order of 10 Mpc give 0.05 G V G 0.2 [11], we face at once the very well-known problem of V. As we will see, and also coherently with what is already known, the value of the cosmological constant requested during inflation to solve the problems of the standard model results much higher than the one requested to fit the present observational data. Thus one has also to claim for a time variable L (see also [12]).

In this paper we analyse some phenomenological aspects connected to the presence of a cosmological constant related to the problem of V and of the time dependence of L in the context of an exactly solved model. We consider a homogeneous and isotropic flat model with an inflationary epoch, a radiation-dominated epoch and a matter-dominated epoch, in which there is a residual cosmological constant. That is we consider the system of equations 2 a O a 1

u

a. a

v

2 4 L 2 pm, (1) r.m1 3 a N a(pm1 rm) 40 , (2)

u

a. a

v

2 4 L 3 1 rm 3 (3)

in which a(t) is the expansion parameter; actually L is considered a function of time, more precisely, as will be shown, it will be considered piecewise constant; rm, pm are

the energy density and the pressure relative to the matter. Equations (1), (3) are the Einstein equations, while (2) is the contracted Bianchi identity. We are using units in which 8 pG 4c41. Finally, considering standard matter, its state equation is, as usual, pm4 (g 2 1 ) rm, with 1 GgG2.

2. – The model

We describe three epochs in which we assume L and g as step-functions, that is L_{(t) 4}./ ´ L1, L2, tiE t E tf, tfE t , (4)

g(t) 4

.

/

´

g14 4 3 , g24 1 , t Eteq, teqE t , (5)

in which L1is the value of the cosmological constant during the inflationary epoch, L2is

the residual value, ti, tfare the initial and final times of the inflationary epoch, teqis the

instant of equivalence between radiation and matter density. We solve (1)-(3) for tiE

Et E tf, tfE t E teq, teqE t and then impose the continuity of the expansion parameter a

and of the total energy density rtot4 L 1 rm in tf, teq; from (1)-(3) these conditions

imply that the Hubble parameter H 4 a.Oa and the total density parameter Vtot4

4rtotO3 H2are continuous at any t [13-15].

To solve (1)-(3), taking into account the state equation, we follow this way: we multiply eq. (3) by a factor bi, then add it to eq. (1), obtaining

2 d dt

u

a. a

v

1 (bi1 3 )

u

a. a

v

2 2 1 3(bi1 3 ) Li2 1 3[bi2 3(gi2 1 ) ] rm4 0 (6)

in which the index i takes into account the different values of g and L in the different epochs, according to (4), (5). Taking bi4 3(gi2 1 ), we obtain a second-order equation

for a(t) in which the term relative to rm does not appear explicitly. Equation (6) is a

Riccati-type equation in a._{Oa , which is possible to solve. We obtain}

a 4aa

k

e( 3 giO2 )kLiO3 t2 cae2( 3 giO2 )kLiO3 t

l

2 O3gi

. (7)

From the (contracted) Bianchi identity, eq. (2), and from the (0, 0) Einstein equation (3) we get ca4 MaO4 Liaa3 gi, where Mais given by Ma4 rma3 gi.

It is interesting to note that the expression (7) for a(t) presents a singularity for any (a, i) if caD 0. This condition is verified, being caconnected with the energy density of

matter. Thus there will be a tssuch that a(ts) 40; it seems thus natural to put the time

origin in ts. Redefining t in such a way that aNt 404 0, the solution takes the form

a 4

.

`

/

`

´

a1

[

e 2kL1O3 t 2 e22kL1O3 t

]

1 O2_, a2

k

e2kL2O3 t2 c2e22kL2O3 t

l

1 O2 , a3

k

e( 3 O2)kL2O3 t2 c3e2( 3 O2)kL2O3 t

l

2 O3 , tiE t E tf, tfE t E teq, teqE t , M1 4 L1a14 4 1 , c24 M2 4 L2a24 , c34 M3 4 L2a33 (8)

in which aa, ca are not the same as given in (7) but they have been opportunely

redefined, by fixing the time origin. Of course, the meaning of ti, tf, teq in (4), (5)

different epochs: rtot4

.

`

/

`

´

L11 M1 a4 , L21 M2 a4 , L21 M3 a3 , tiE t E tf, tfE t E teq, teqE t . (9)

As we have said above, we impose the continuity conditions aNtf24 aNtf1, rtotNtf24 rtotNtf1,

(10)

aNteq24 aNteq1, rtotNteq24 rtotNteq1.

(11)

In this way, from (3) and from the definition of V, we have that H(t) and V(t) are continuous at any t.

It is noteworthy that, assuming (4), (5) and taking into account (9), we introduce a discontinuity, at the instants tf, teq, in the equation of state [13-15] for the total energy

density ptot4 (gtot2 1 ) rtot, being gtot4 g(t) rmO

(

L(t) 1rm

)

; then the entropy S and

the scalar curvature R are discontinuous too [13-15] in the same instants, being, respectively, S 4 a 3_g tot T ; R 42 6

g

aO a 1 H 2

h

_, (12)

and considering also that (1) can be written as

2

g

a O a

h

1

u

a. a

v

2 4 2 (gtot2 1 ) rtot. (13)

Thus the entropy production in tftakes place through a phase transition and takes into

account the production of matter at the end of the inflationary epoch through the condition (10) on rtot. In this way, the production of matter through phase transition

can be directly connected with the decaying of L and thus with its time variability. There is a further remark we would like to do: in the context we are considering, that is the one of minimal coupled theory without scalar field, the only allowed behavior of the “cosmological constant” with time is the above-considered step-function, in order to have compatibility with the Bianchi contracted identity everywhere but in the instants of discontinuity.

Referring to the first of (9), we can consider the beginning of inflationary epoch tias

the instant in which there is equivalence between the energy density relative to the cosmological constant and the one relative to the matter, that is we define tiin such a

way that

L14

M1

a4

N

ti

so that, for t b ti, one would have a radiation-dominated pre-inflation epoch and for

t c tione has the inflationary epoch dominated by the cosmological constant. Thus

ti4 1 2

o

3 L1 ln ( 1 1k2) . (15)

To solve the problems of the standard model (horizon, flatness, entropy), the number of e-folding during inflation has to be Ne-foldingF 67 [9, 10]. From this condition, assuming

that kL1O3 t c 1 during inflation, which seems to be very reasonable, we get a

condition on L1and tf, given by L1tf2F 1.4 Q 104. The validity of the model is assumed, of

course, for t FtP, with tPC 5.4 Q 10244s the Planckian time. Imposing that, at tP, the total

energy density is just the Planck density, we obtain an estimate for L1. Thus we

write M1 a4

N

tP 4 rP (16)

in which rP is the Planckian density. If we assume that at tP the radiation dominates,

that is tPbti, we can develop the first of (7) in kL1O3 t to the first order

(

radiation

behavior for a(t)

)

, obtaining at once an estimate for L1, which comes out to be negative.

This means that we have to take into account also the cosmological-constant term. Assuming in the first approximation that it dominates, we get a lower bound for L1,

that is L1F 8.4 Q 1087s22. From (15) one gets tiG 8.3 Q 10245 s, which is less than the

Planckian time. This only means that our model can be considered reasonable starting with a cosmological-constant–dominated era.

The continuity conditions (10), (11) give some relations between the constant aa, ca

present in eq. (8); taking into account the relation between L1, tf we get

a14

y

2 1 1kL12

z

1 O2 _a 0ekL2O3 (tf2 t0)[ 1 2r12e4kL2O3 (tf2 teq)]1 O6 ekL1O3 tf_{[ 1 2r} 12ekL2O3 ( 4 tf2 teq2 3 t0)]2 O3 , (17) a24 a0[ 1 2r12e4kL2O3 (tf2 teq)]1 O6 ekL2O3 t0_{[ 1 2r} 12ekL2O3 ( 4 tf2 teq2 3 t0)]2 O3 , (18) a34 a0 ekL2O3 t0_{[ 1 2r} 12ekL2O3 ( 4 tf2 teq2 3 t0)]2 O3 , (19) c24 r12e4kL2O3 tf, c34 r12ekL2O3 ( 4 tf2 teq) (20)

in which t0is the present age,

a04 aNt0, L124

o

L1 L2 and r12C 21 1kL1OL2 1 1kL1OL2 .

A quite natural assumption is L1D L2; it can be seen that this is equivalent to have

3. – Compatibility with observations

Now we are going to compare the theoretical values we obtain from this simple model with some of the most recent observational cosmological quantities. The comparison with the experimental data can be done through the constants M2, M3,

which can be connected with the energy density of the matter and of the radiation at the present age, which we call, respectively, m0, e0.

Assuming, as usual, e0bm0, one can write

M24 e0a04, M34 m0a03.

(21)

Using the relations existing between c2, M2and c3, M3as given by (8) and taking into

account eqs. (17)-(20), we get

e04 4 L2c2a24 a04 4 4 L2r12e kL2O3 (tf2 t0)_{[ 1 2r} 12e4kL2O3 (tf2 teq)]2 O3 [ 1 2r12ekL2O3 ( 4 tf2 teq2 3 t0)]8 O3 , (22) m04 4 L2c3a33 a03 4 4 L2r12e kL2O3( 4 tf2 teq2 3 t0) [ 1 2r12ekL2O3( 4 tf2 teq2 3 t0)]2 . (23)

Being tf, teqbt0, we also get an expression for the present age:

t04 1 H0



1 Q dx x

k

Vm0x 3 1 VL0 (24)

in which H0, Vm0, VL0 are, respectively, the Hubble parameter, the contribution of

matter and cosmological constant to V at the present age. The observational value of e0

comes from the black-body law

e04 sBT04C 4.649 Q 10234 g cm23

(25)

in which sB is the radiation-density constant and T04 2.726 6 0.010 K is the CMBR

temperature [16].

The most recent estimates of the Hubble parameter H0come from various distances

calibrators on different scales and are (see [2, 6, 7, 17])

H04 100 h km s21 Mpc21, 0.55 GhG0.85 .

(26)

The most recent estimates of Vm0come from the comparison of CDM models vs. redshift

surveys and from studies on the dynamics of cosmic flows; they give (see [18-20]) Vm0F

F0.2–0.3 at 2 s level or more. Thus, from the definition of V, the model being flat, we get m0 , est4 3 H02Vm04 1.879 Q 10 229_h2_V m0 g cm 23_, (27) L2 , est4 3 H02VL04 3.151 Q 10 235_h2_V L0 s 22_. (28)

We obtain an estimate of tfconsidering that, for tfE t E teq, the matter is prevalently

ultrarelativistic, thus one has the relation between the energy density rm and the

absolute temperature T rm4 sB 2 g(T) T 4 (29)

in which g(T) are the effective spin degree of freedom. Considering the epoch just after tf, the radiation dominates on the cosmological constant and, being the hypothesis of an

efficient reheating verified by construction, by equating the expression of rm(tf1) with

rm(Tf1), one gets tf in terms of Tf1, that is, the right limit on Tf. Such a quantity is

constrained by the bariogenesis [21], being Tf1F 1010GeV, which implies a constraint

on tf, that is tfG 3.7 Q 10229 s and this is compatible with the relation between L1, tfand

with the constraint we found on L1.

Assuming tf4 10229, teq4 1012, Vm04 0.3, compatible with all the considerations we

have done, we find that the values of m0, e0 given from the model are substantially

compatible with those coming from the observations. In particular, we find that the values of e0 , mod obtained from the model, see (22), with h varying according to (26),

differ from the values e0 , estgiven from (25) as

e0 , est2 e0 , mod

e0 , est

N

h 40.554 0.43,

e0 , est2 e0 , mod

e0 , est

N

h 40.854 2 0.83 ,

(30)

essentially unaffected by T0 varying within the experimental errors and for increasing

Vm0. This means that there is full compatibility between the values given from the

model and the observational data; the value of h which minimizes the square of the relative difference for e0to a value significantly lower than 1024 is given by h 40.6777,

which is compatible with the estimate given by (26).

The values of m0 , modobtained from the model, see (23), with h varying according to

(26), differ from the values m0 , estgiven from (27) as

m0 , est2 m0 , mod m0 , est

N

h 40.554 1.8 Q 10 26_, m0 , est2 m0 , mod m0 , est

N

h 40.854 2.2 Q 10 26_. (31)

In this case we do not find a full compatibility, but the relative difference is of one part over 106_{. The agreement increases with increasing V}

m0and decreasing h.

The values of the equivalence temperature, for h 40.6777, are found to be TeqNT04 2.7164 1.303 eV, TeqNT04 2.7364 1.313 eV, which are just one order of magnitude

less than the decoupling temperature, given by Tdec4 13 eV, and for the temperature

immediately after inflation Tf1NT04 2.7164 5.041 Q 10

18_{GeV, T}

f1NT04 2.7364 5.078 Q 10 18_GeV,

which are compatible with the constraint imposed by the bariogenesis.

Moreover, we find for the present age the value t0 , mod4 13.9 Q 109y, for h 40.6777,

compatible with the most recent estimates which give t0 , est4 ( 14 6 2 ) Q 109y [7]. Of

course the agreement decreases with increasing h and increases with decreasing Vm0.

Finally, we want to stress that the behavior of a(t) given from (8) seems to be quite different from the standard expansion a Pt2 O3 _{of the universe filled of dust, as our}

present universe appears. One gets the standard expansion from (8) ifkL2O3 t b 1; this

turns out to be no longer strictly verified at the present age, since we getkL2O3 t0C

C0.7. This means that the effects of the cosmological-constant term on the expansion are no longer negligible. This is something which could be taken into account in the observational measurements.

4. – Conclusions

We have constructed a phenomenological model in which the cosmological constant is imposed (by hands) to be a step-function depending on the epoch. Also ordinary standard matter is taken into consideration as radiation at the beginning and as dust after the equivalence. The model is exactly solvable and allows to implement an inflationary epoch after which we found substantial agreement with the most recent observational data concerning the values of V, the age of the universe, the CMBR temperature at equivalence and today. The construction can be perfectly compatible with such models which call for an amount of barionic matter, cold dark matter and cosmological constant in order to explain cosmological dynamics and large-scale structure formation after an inflationary expansion [22]. Although this is a toy model in which a sharp transition between the two values of L is invoked, it justifies how a time-dependent cosmological constant could affect early and present cosmological dynamics and, in some sense, be in agreement with observational data. This analysis confirms in any case how important it could be to specify the concept of “cosmological constant” in a wider way and in a more general context, such as, for example, that of nonminimally coupled scalar-tensor theories, where also scalar field(s) can be con-sidered in dynamics and, in general, be nonminimally coupled with geometry [23]. As concluding remark we consider more carefully the non-completely satisfactory compatibility expressed by (31); actually, giving h the value h 40.7 compatible with the data, the square of the relative difference between m0 , est and m0 , mod reaches its

minimum for a negative value of teq. Thus the transition from radiation to matter would

occur before the initial singularity. This means that to have full compatibility between the values of the model and the observational data relatively to the matter energy density, a radiation-dominated era does not take place. This suggests an explanation of the disagreement concerning the matter in the presence of radiation, because we have totally neglected its presence after teq, but we have taken it into account in the

comparison with the data. The order of magnitude of the disagreement is in fact less than the order of magnitude of the ratio between the present energy density of the radiation and that of the matter

(

see eqs. (31) and (25), (27)

)

. It is reasonable to think, in order to solve these discrepancies, that a model with non-zero initial conditions on the scalar field (see [24]) can be revisited with the phenomenological approach used in this paper. This will be one of the further developments of our future research.

* * *

One of us (AAM) would like to thank S. MATARRESE for the useful discussions on this topic.

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