Cosmological Constant or
Cosmological Potential P. Fiziev
Theoretical Physics Department Sofia University
[email protected] August 24, 2000
At present GR is: A SUCCESSFUL THEORY of gravity in description of gravitational phenomena at: • laboratory and earth surface scales, • solar system and star systems scales.
QUITE GOOD in description of these phenomena: • at galaxies scales, and • at the scales of the whole Universe. PROBLEMATIC in description of: • rotation of galaxies, • initial singularity problem, early Universe, • recently discovered accelerated expansion of the Universe, • vacuum energy problem.
The most promising modern theories like supergravity and (super)string theories incorporate naturally GR but at present: • are not developed enough to allow real experimental test, • introduce large number of new fields without real physical basis. Therefore it seems meaningful to look for some minimal extension of GR which: • is compatible with known gravitational experiments, • promises to overcome at least some of the problems, • may be considered as a part of more general modern theories.
In this talk we outline the general properties of such model with one additional scalar field Φ which differs from known inflationary models. We call MINIMAL DILATONIC GRAVITY (MDG) the scalar-tensor model of gravity with action
c AG,Λ = − 2¯ κ
Z
4
q
d x |g|Φ R + 2ΛΠ(Φ) ,
(1)
• Branse-Dicke parameter ω(Φ) ≡ 0 (i.e., without standard kinetic term for Φ !), • cosmological constant Λ and • dimensionless cosmological factor Π(Φ). The matter action AM and matter equations of motion will have usual GR form.
Equations for metric gαβ and dilaton field Φ:
u)Φ = c¯κ Tαβ , Φ Gαβ −ΛΠ(Φ)gαβ −(∇α∇β −gαβt 2
dV (Φ) = 3c¯κ T. t uΦ+Λ dΦ
(2)
2
yield energy-momentum conservation law: ∇α Tβα = 0 In addition we have the important relation:
dΠ (Φ) + Π(Φ) = 0. R + 2Λ Φ dΦ
(3)
In (2) the quantity: dV dΦ
dΠ = 23 Φ Φ dΦ −Π
introduces dilatonic potential V (Φ). It is convenient to introduce, too, a cosmological potential: U (Φ) = ΦΠ(Φ).
The investigation of MDG was started by O’Hanlon (1972) in connection with Fujii’s theory of massive dilaton, but without any relation with cosmological constant problem. In our special scalar-tensor model of gravity the cosmological factor Π(Φ) (or the cosmological potential U (Φ)) is the only unknown function which has to be chosen: • to comply with all gravitational experiments and observations • to solve the following inverse cosmological problem: Determination of the factor Π(Φ) which yields given time evolution of the scale parameter A(t) in Robertson-Walker (RW) model of Universe.
The action (1) of MDG is a Helmholz action of nonlinear gravity with lagrangian LN LG ∼
q
|g|f (R),
or a (4d) low energy limit of superstringy action (for metric and dilaton only) in some new frame. It follows from stringy lagrangian LString ∼
q
−2φ
|g|e
2
R + 4(∂φ) + VSU SY (φ) after transformation (in D-dimensions): √ gµν → e±4φ/ (D−2)(D−1)gµν ,
φ → Φ = exp −2φ 1 ±
q
(D − 2)/(D − 1)
.
The potential VSU SY (φ) may originate from SUSY breaking (due to gaugino condensation, or in a more complicated way). At present its form is not known exactly. The action (1) appears, too, in a new model of gravity with torsion and unusual local conformal symmetry after its breaking in metricdilaton sector only (PF: gr-qc/9809001).
An essential new element of our MDG (PF: gr-qc/9911037) is the nonzero cosmological constant Λ. The astrophysical data: ΩΛ = .65 ± .13,
H0 = (65 ± 5) km s−1M ps−1
give observed value Λobs = 3ΩΛH02c−2 = (.98 ± .34) × 10−56 cm−2. We accept this value as a basic quantity which defines natural units for all other cosmological quantities: • cosmological length: q
Ac = 1/ Λobs = (1.02 ± .18) × 1028 cm, • cosmological time: Tc := Ac/c = (3.4±.6)×1017 s = (10.8±1.9)Gyr,
• cosmological energy density: Λc2 = (1.16 ± .41) × 10−7g cm−1 s−2, εc := κ • cosmological energy: −1/2 2 −1 c κ = (3.7±.7)×1077 erg, Ec := 3A3 c εc = 3Λ
• cosmological momentum: q
Pc := 3c/(κ Λobs) = (1.2 ± .2) × 1067 g cm s−1 • and cosmological unit for action: h, Ac := 3c/(κΛobs) = (1.2 ± .4) × 10122 ¯ κ being Einstein constant. Further we use dimensionless variables like: τ := t/Tc, a := A/Ac, h := H Tc (H := A−1dA/dt being Hubble parameter), c = εc/|εc| = ±1, := ε/|εc|-matter energy density, etc.
Solar System and Earth-Surface Gravitational Experiments Properties of cosmological factor Π(Φ) derived from known gravitational experiments: 1.The MDG with Λ = 0 contradicts to solar system gravitational experiments. The cosmological term ΛΠ(Φ) 6= 0 in action (1) is needed to overcome this problem. 2. In contrast to O’Hanlon’s model we wish MDG to reproduce GR with Λ 6= 0 for some ¯ = const 6= 0. Φ=Φ Then from action (1) we obtain normalization condition for cosmological factor ¯ =1 Π(Φ) and Einstein constant ¯ κ=¯ κ/Φ.
3. In vacuum, far from matter MDG have to allow week field approximation (|ζ| 1): ¯ + ζ). Φ = Φ(1 Then the linearized second of equations (2): 2 = 3cκ T t uζ + ζ/lΦ 2
gives dΠ ¯ = 1/Φ. ¯ (Φ) dΦ Taylor series expansion of the function ¯ gives relation around the value Φ
dV dΦ
(Φ)
d2Π ¯ = 32 p−2Φ ¯ −2. ( Φ) dΦ2 Then 3 2 Π = 1+ ζ + 2 ζ + O(ζ 3), 4p lΦ p= Ac being dimensionless Compton length of dilaton in cosmological units.
4. Point particles of masses ma as source of metric and dilaton fields give in Newtonian approximation gravitational potential ϕ(r) and dilaton field Φ(r):
2
ϕ(r)/c = −
G c2
X a
ma |r−ra |
1+α(p)e
− p2 1 6
¯ =1 + Φ(r)/Φ
−|r−ra |/lΦ
X a
2 G 3 c2 (1− 4 p2 ) 3
ma M
X a
(|r − ra|/lΦ)2 ,
(4)
e−|r−ra|/lΦ ,
(5)
ma |r−ra |
4 p2) is Newton constant, M = (1− G = κc 8π 3 The term 2
2
− p 1 6
X a
ma M
2
(|r − ra|/lΦ) = − Λ|r − 1 6
X a
ma M
P a ma .
ra|2+const
in ϕ is known from GR with Λ 6= 0. It represents an universal anty-gravitational interaction of test mass m with any mass ma via repellent elastic force
FΛ a = 13 Λmc2 mM (r − ra). a
(6)
For solar system distances l ≤ 1000AU neglecting the Λ term ( of order ≤ 10−24 ) we compare the gravitational potential ϕ with specific MDG coefficient 1 + 4p2 α(p) = 3 − 4p2 with Cavendish type experiments and obtain an experimental constraint lΦ ≤ 1.6 [mm], or p < 2 × 10−29 . Hence, in the solar system phenomena the factor e−l/lΦ has a fantastic small values (< exp(−1013) for the Earth-Sun distances l, or < exp(−3 × 1010 ) for the Earth-Moon distances l) and there is no hope to find some differences between MDG and GR in this domain. The constraint mΦc2 ≥ 10−4[eV ] does not exclude a small value (with respect to the elementary particles scales) for the rest energy of hypothetical Φ-particle.
5. The parameterized-post-Newtonian (PPN) solution of equation (2) is complicated, but because of the constraint p < 10−28 we may use with great precision Helbig’s PPN formalism (for α = 13 ). Because of the condition ω ≡ 0 we obtain much more definite predictions then usual general relations between α and the length lΦ: • Nordtvedt Effect: In MDG a body with significant gravitational P m will not move along self-energy EG = b6=c G |rm−r | geodesics due to additional universal antygravitational force: b
b
c
c
FN = − 23 EG ∇Φ.
(7)
For usual bodies it is too small even at distances |r − ra| ≤ lΦ, because of the small factor EG. Hence, in MDG we have no strict strong equivalence principle nevertheless the week equivalence principle is not violated.
The experimental data for Nordtvedt effect caused by the Sun are formulated as a constraint η = 0 ± .0015 on the parameter η which in MDG becomes a function of the distance l to the source: η(l) = − 12 (1 + l/lΦ) e−l/lΦ . This gives constraint lΦ ≤ 2 × 1010 [m]. • Time Delay of Electromagnetic Waves The action of electromagnetic field does not depend on the field Φ. Therefore influence of Φ on the electromagnetic waves in vacuum is possible only via influence of Φ on the space-time metric. The solar system measurements of the time delay of the electromagnetic pulses give the value γ = 1 ± .001 of this post Newtonian parameter. In MDG this yields the relation (1 ± .001)g(lAU ) = 1 and gives once more the constraint lΦ ≤ 2 × 1010 [m]. Here g(l) := 1 + 13 (1 + l/lΦ)e−l/lΦ .
• Perihelion Shift For the perihelion shift of a planet orbiting around the Sun (with mass M ) in MDG we have: k(lp ) δϕGR . δϕ = g(lp ) Here lp is the semimajor axis of the orbit of planet and k(lp ) ≈ 1 +
1 18
4+
lp2 lp c2 2 lΦ GM
1 −2lp /lΦ e−lp/lΦ − 27 e
is obtained neglecting its eccentricity. The observed value of perihelion shift of Mercury gives the constraint lΦ ≤ 109[m]. Conclusion: In presence of dilaton field Φ are impossible essential deviations from GR in solar system. Observable deviations from Newton law of gravity may not be expected at distances greater then few mm.
Vacuum Energy and True Vacuum Solution in MDG Total (true) tensor of energy momentum: T Tµν := Tµν + < ρ0 > c2gµν ,
(8)
< ρ0 > being averaged density of zero quantum fluctuations. For true vacuum solution of MDG: Φ = Φ0 = const, gµν = ηµν from dynamical equations (2) we obtain: Φ0
dΠ (Φ0) + Π(Φ0) = 0 dΠ
(9)
(10)
c2 0 0 − < ρ > c2g , (11) Tµν = − ΛU0gµν = T Tµν µν 0
¯ κ where U0 = Φ0Π(Φ0) = Φ0Π0. But for true vacuum solution 0 ≡ 0. T Tµν
(12)
This way we obtain < ρ0 >=
1 1 ΛU0 = ΛΠ0 ¯ κ κ
(13)
Hence in MDG:
True Vacuum
⇒
Minkowski Space-Time:
T Tµν ≡ 0.
Physical Vacuum
⇒
de Sitter Space-Time:
T Tµν =< ρ0 > c2gµν .
– a physically sound picture !
The real word looks like de Sitter Universe created by zero quantum vacuum fluctuations and perturbed by other matter and radiation fields.
For < ρ0 > calculated using Plank length as a quantum cuttoff the observed value of Λ gives:
¯ = κ < ρ > /Λ ≈ 10118 U (Φ0)/U (Φ) and causes the famous cosmological constant problem in standard theory. We see that: • It is obviously close in order to the ratio of h: cosmological action Ac and Planck constant ¯ ¯ ≈ 10−4Ac/¯ h. U (Φ0)/U (Φ) • In MDG there is no crisis caused by this big number, because it gives ratio of the values of cosmological potential for different solutions: ¯ i.e. in different universes. Φ0 and Φ,
If we calculate the values c U0 Voll = −Λ A0 G,Λ ¯ κ and c ¯ Voll ¯G,Λ = Λ U A ¯ κ of the very action (1) and introduce correc sponding specific actions α0 = −Λ ¯ κ U0 and cU ¯ α ¯=Λ¯ κ , i.e., actions per unit volume, we can rewrite the above observed result in a form:
h/Ac = |α0| × 10−118 . α ¯ ≈ −α0 × 104 ¯ One can hope that such new and quite radically changed formulation of the cosmological constant problem in MDG will bring us to its resolution. For example it’s easy to think that this results is determined by evolution of the Universe.
Application of MDG in Cosmology Consider RW adiabatic homogeneous isotropic Universe with 2 2 2 2 ds2 RW = c dt − A dlk ,
t = Tcτ , A(t) = Aca(τ ) and dimensionless dl2 2 2 2 2 + l (dθ + sin θ)dϕ 1 − kl2 (k = −1, 0, 1) in presence of matter with energy¯ and pressure p = εcp(a)/Φ. ¯ density ε = εc(a)/Φ dlk2 =
Basic dynamical equations of MDG for RW Universe are: 1 d2a a dτ 2 1 da dΦ a dτ dτ
da 2 + a1 ( dτ ) + ak =
+Φ
2
1 a2
2
da 2 dτ
( ) +
k a2
1 3
dΠ Φ dΦ (Φ) + Π(Φ) ,
= 13 (ΦΠ(Φ) + (a)) (. 14)
da (τ (a)) The use of Hubble parameter h(a) = a−1 dτ (τ (a) – inverse function of a(τ )), new variable
λ = ln a and prime for differentiation with respect to λ gives the system for Φ(λ) and h2(λ): 1 2
2 0
(h ) + 2h + ke−2λ = 2 0 2 −2λ 2
dΠ dΦ
Φ (Φ) + Π(Φ) , Φ = 13 (ΦΠ(Φ) + (eλ)).
h Φ + h + ke and relation
1 3
Z a
τ (a) =
ain
da/(a h(a)) + τin.
Excluding cosmological factor Π(Φ) we have:
Φ00 +
h0 h
−1 Φ0 +2
h0 h
−2 −2λ − kh e Φ = 3h1 2 0.(15) q
In terms of the function ψ(a) = it reads:
|h(a)|/a Φ(a)
ψ 00 + n2ψ = δ,
(16)
where we introduce new functions −n2 =
1 h00 2h
−2λ − 14 ( hh )2 − 52 hh + 14 + 2k e , h 0
0
q
δ=
1 3
2
d a/|h|3 da .
(17)
Now we are ready to consider The inverse cosmological problem: to find a cosmological factor Π(Φ) (or potentials V (Φ), or U (Φ)) which yield given evolution of the Universe, determined by function a(τ ). A remarkable property of MDG: An unique solution of this problem exist for almost any three times differentiable function a(τ ).
Indeed: for given a(τ ) construct a function h(λ) and find the general solution Φ(λ, C1, C2) of the linear second order differential equation (15). The two constants C1,2 have to be determined from the additional conditions ¯ = 1, Π(Φ)
dΠ dΦ
¯ =Φ ¯ −1, (Φ)
d2 Π dΦ2
¯ = 32 p−2Φ ¯ −2. (Φ)
–self-consistence conditions at point ¯ λ which is real solution of the algebraic equation r(¯ λ) = −4, r(λ) = −6
1 2
2 0
2
−2λ
(h ) + 2h + ke
being dimensionless scalar curvature: r = R/Λ. Then:
2 0 2 2 2 0 ¯ = −4¯ h ¯ r , j00(1+ 43 p ) + 4p ¯ Φ 1+ 43 p / ¯ 0 2 0 2 ¯ = −1p ¯ ¯ /Φ r / 1+ 4 p . (18) Φ 3
3
Here j00 = G00/Λ = 3 h2 + ke−2λ
is dimensionless 00-component of Einstein tensor. Hence, the values of all ”bar” quantities (including ¯ κ
in action (1)) may be determined from time evolution a(τ ) of the Universe via the solution ¯ ¯ λ = ln ¯ a of the equation (3). In their turn Φ ¯ 0 determine the values of constants C1,2 and Φ and an unique solution Φ(λ) of the equation (15): Φ(λ) = C1Φ1(λ) + C2Φ2(λ) + Φ(λ) where Φ1(λ) and Φ2(λ) are a fundamental system of solutions of the homogeneous equation associated with non-homogeneous one (15). Then Z λ
Z λ
!
¯ a Φ1 Φ2 − Φ1 d Φ2 d . Φ = ¯ ¯ ¯ (3¯ h∆) ah ah λ λ where ∆(λ) = Φ1Φ02 − Φ2Φ01. The cosmological factor Π and the potential V as functions of the variable λ are determined by equations Π(λ) = j00 + 3h2Φ0/Φ − /Φ, Z V (λ) =
2 3
0
0
Φ ΦΠ − Φ Π dλ
(19)
which define functions Π(Φ) and V (Φ) implicitly, too.
Simple exactly soluble examples: 1. Evolution law a(τ ) = (ωτ )1/ν , (ω is free parameter) gives h(λ) = ων e−νλ, −n2(λ) =
1+10ν+ν 2 4
+ 2k ων e2(ν−1)λ, 2
2
2(1−ν) = k for ¯ and the equation ¯ a2 + 3(ν−2)ω ¯ a a. 2ν 2
2
i) For ν ≥ 2 we have real solution ¯ a only if k = +1: Φ1(a) = √
µ :=
ν+1 ν−1 a 2 Iµ ba , Φ2(a)
1+10ν+ν 2 2(ν−1)
=
ν+1 ν−1 a 2 Kµ ba ,
being the order of Bessel funcq
2
2ν . For GRtions Iµ, Jµ, Kµ , Yµ and b := (ν−1) ω √ like law a ∼ τ (ν = 2) in MDG we obtain positive value (k = +1) for three-space curvature, ¯ λ = 0, µ = 52 and Bessel functions are reduced to elementary functions. 2
2
ii) When ν < 2 all values k = −1, 0, +1 are admissible:
- for k = +1 the solutions Φ1,2 are the same as above; - for k = 0 we have Φ1,2 =
ν+1 a 2 ±µ(ν−1);
- for k = −1 the solutions are:
ν+1 ν+1 ν−1 ν−1 , Φ2 = a 2 Yµ ba . Φ1 = a 2 Jµ ba
In the special case of linear evolution a ∼ τ , (ν = 1) −n2(λ) = 3 + 2k ω √ 1± −n2 Φ1,2(a) = a 2
and the root ¯ λ = 12 ln 32 (k + ω 2) is real for all values of ω 2 > 0, if k = 0, +1. For k = −1 the root ¯ λ will be real if |ω| > 1.
2. Evolution law a(τ ) = eωτ gives −2λ e h(λ) = ω, −n2(λ) = 14 + 2k ω 2
2) = k with root ¯ 2 2 and the equation ¯ a (1−3ω a= 3 q 3 . Now we have the following solutions: 2|1−3ω | 2
i) |ω| < k = +1; ii)|ω| > k = −1.
√ 3 3
√
3 3
√
: Φ1 = a cosh
2 |ω|a
√
: Φ1 = a cos
2 |ω|a
√
, Φ2 = a sinh
, Φ2 = a sin
2 |ω|a
√ 2 |ω|a
,
,
Conditions r(¯ λ) = −4 and (18) exclude exact exponential expansion of spatially flat Universe (k = 0 ). For inflationary scenario in this case one may use a scale factor a(τ ) = eωτ + const which turns to be possible if const 6= 0.
Specific properties of MDG: 1) If n > 0 dilatonic field Φ(a) oscillates; if n < 0 such oscillations do not exist. 2) Dilatonic field Φ may change its sign, i.e. phase transitions of the Universe from gravity (Φ > 0) to anty-gravity (Φ < 0) and viceversa are possible in general. 3) In spirit of Max principle Newton constant ¯∼ depends on presence of matter: G ∼ 1/Φ 1/¯ . 4) For simple functions a(τ ) the cosmological factor Π(Φ) and potentials V (Φ) and U (Φ) may show unexpected catastrophic behavior: ∼ (∆Φ)3/2 (∆Φ = Φ − Φ(λ?)) in vicinity of the critical points λ?: Φ0(λ?) = 0 of the projection of analytical curve {Π(λ), Φ(λ), λ} on the plain {Π, Φ}.
Scale factors a(τ ) exist yielding an everywhere analytical cosmological factor Π(Φ) and potentials V (Φ) and V (Φ), too. 5) Clearly one can construct MDG model of Universe without initial singularities: a(τ0) = 0 (typical for GR) and with any desired kind of inflation. 6) Because the dilaton field Φ is quite massive, in it will be stored significant amount of energy. An interesting open question is: may the field Φ play the role of dark matter in the Universe? A very important problem is to reconstruct the cosmological factor Π(Φ) of real Universe using proper experimental data and astrophysical observations. GENERAL CONCLUSIONS:
• MDG offers new curious possibilities which deserve further careful investigation. • Instead of Cosmological Constant we have to prefer a Cosmological Potential, because it yields much richer theory which is more suitable to describe the real world.
