научная статья по теме TOWARDS A SOLUTION OF THE COSMOLOGICAL CONSTANT PROBLEM Физика

Текст научной статьи на тему «TOWARDS A SOLUTION OF THE COSMOLOGICAL CONSTANT PROBLEM»

Pis'ma v ZhETF, vol.91, iss.6, pp. 279-285

© 2010 March 25

Towards a solution of the cosmological constant problem

F. R. Klinkhamer*1'), G. E. Volovik *+ * Institute for Theoretical Physics, University of Karlsruhe, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany * Low Temperature Laboratory, Aalto University, FI-00076 AALTO, Finland + Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia

Submitted 25 January 2010 Resubmitted 8 February 2010

The standard model of elementary particle physics and the theory of general relativity can be extended by the introduction of a vacuum variable which is responsible for the near vanishing of the present cosmological constant (vacuum energy density). The explicit realization of this vacuum variable can be via a three-form gauge field, an aether-type velocity field, or any other field appropriate for the description of the equilibrium state corresponding to the Lorentz-invariant quantum vacuum. The extended theory has, without fine-tuning, a Minkowski-type solution of the field equations with spacetime-independent fields and provides, therefore, a possible solution of the main cosmological constant problem.

1. Introduction. The main cosmological constant problem is to understand why, naturally, the quantum-mechanical zero-point energy of the vacuum does not produce a large cosmological constant or, in other words, to discover the way the zero-point energy is canceled without fine-tuning the theory. Restricting to established physics, this problem was formulated by Weinberg in the following pragmatic way [1, 2]: how to find an extension of the standard model of elementary particle physics and the theory of general relativity, for which there exists, without fine-tuning, a Minkowski-spacetime solution with spacetime-independent fields.

An adjustment-type solution of the cosmological constant problem appears, however, to be impossible with a fundamental scalar field and Weinberg writes in the last sentence of Sec. 2 in Ref. [2] that, to the best of his knowledge, "no one has found a way out of this impasse." In this Letter, we present a way around the impasse, which employs a quantity q that acts as a self-adjusting scalar field but is non-fundamental [3-5].

The main goal of the present publication is to describe, in a more or less consistent way, a particular theoretical framework for addressing the cosmological constant problem. Obviously, this builds on previous work of the present authors and many others (see citations below). But there are also two important new results, which will be indicated explicitly.

2. Minkowski equilibrium vacuum. Our discussion starts from the theory outlined in Ref. [4]. We introduce a special quantity, the vacuum "charge" q, to describe the statics of the quantum vacuum. A concrete

^e-mail: frans.kKnkhamer0kit.edu; volovik0boojum.hut.fi

example of this vacuum variable is given by the four-form field strength [6-14] expressed in terms of q as Fa/3-yS = <7 — detgeap7$ (see below for further details). This particular vacuum variable q is associated with an energy scale E\jy that is assumed to be much larger than the electroweak energy scale Eew ~ 103 GeV and possibly to be of the order of the gravitational energy scale -Epianck = « 2.44-1018 GeV. Here, and in the

following, natural units are used with % = c = 1.

Specifically, the effective action of our theory is given

by

Seff[A g,tl>] = - [ d4x y^drt^ (K(q) R[g] + J R4 V

+e(q) + £!mM), (la)

Fa/3-fS = qtapiS \/-det g = V^A^^ , (lb)

<f (lc)

where R denotes the Ricci curvature scalar, eQja7i the Levi-Civita tensor density, VQ the covariant derivative, and the square bracket around spacetime indices complete anti-symmetrization. Throughout, we use the same conventions as in Ref. [1], in particular, those for

the Riemann curvature tensor and the metric signature (- + ++)•

The vacuum energy density c in (la) depends on the vacuum variable q = q[A, g] and the same is assumed to hold for the gravitational coupling parameter K. The single field \j) combines all the fields of the standard model (spinor, gauge, Higgs, and ghost fields [15]) and, for simplicity, the scalar Lagrange density in (la) is taken to be without direct q dependence. The orig-

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inal standard model fields collected in tfi(x) are quantum fields with vanishing vacuum expectation values in Minkowski spacetime (this holds, in particular, for the physical Higgs field H(x) [15]). The effective action takes ij)(x) to be a classical field, but has additional terms to reflect the quantum effects [16]. The metric field gapix) and the three-form gauge field Ag7,$(») [or other ^-related fields discussed later on] are, for the moment, considered to be genuine classical fields.

The setup, now, is such that a possible constant term Asm in £|m (which includes the zero-point energies from the standard model fields) has been absorbed in e(q), so that, in the end, ^ImWns] contains only i/i-dependent terms, with the metric gap (or vierbein eentering through the usual covariant derivatives. In short, the following holds true:

£sm №>,»?] = 0,

(2)

where Vo denotes the constant values for the standard model fields over Minkowski spacetime and rj stands for the Minkowski metric r)ap = diag( —1, 1, 1, 1) in standard coordinates.

The actual spectrum of the vacuum energy density (meaning the different contributions to c from different energy scales) is not important for the cancellation mechanism to be discussed in this Letter. Still, we assume, for definiteness, that the vacuum energy density e(q) splits into a constant part and a variable part:

«(?) = Abare + eVar (q) = Asm + Auv + ew(?), (3)

with demi/dq ^ 0, a constant term Asm of typical size |Asm| ~ (Eew)4 removed from according to (2), and a possible extra contribution Auv of size |Auv| ~ ~ (Euv)4 from the unknown physics beyond the standard model. For definiteness, we also assume that evar (q) contains only even powers of q and recall that q2 is defined by (lc) in terms of the three-form gauge field A entering the field strength (lb). Allowing for a general even function e(q) instead of the single Maxwell-type term | q2 considered in the previous literature [6 -10] will turn out to be an important ingredient for the cancellation of Abare values of arbitrary sign.

The generalized Maxwell and Einstein equations from action (la) have been derived in Ref. [4]. The generalized Maxwell equation reads

r, ( - Fa^s fde(q) dK(q)\\ n/i.

and reproduces the known equation [6, 7] for the special case e(q) = | q2 and dK/dq = 0. The first integral of (4) with integration constant /i and the final version of

the generalized Einstein equation then give the following generic equations [4]:

de(q) dq

2 К (Raf}

dK(q)

■ gaf} R/2) = ^2 (va V^ - gaf} □) K(q)

(e(q) -nq)g'

.ap

rp(X0 ' JSM'

(5a)

(5b)

where is the energy-momentum tensor corresponding to the effective Lagrangian appearing in (la) and (2). From general coordinate invariance, the energy-momentum tensor is known to have a vanishing covariant divergence, VQ = 0.

For the special case K(q) = K0 = const, (5b) reduces to the standard Einstein equation of general relativity. For the general case dK/dq ^ 0, the action (la) and the resulting field equation (5b) correspond to those of Brans-Dicke theory [17], but without kinetic term for the scalar degree of freedom (wbd = 0). See also the related work on inflation theory [18], dark-energy models [19 - 22] and the connection to g-theory [23 - 25].

The crucial difference between our theory and conventional f(R) modified-gravity theories [18-22] lies in the appearance, for us, of the integration constant /i after integration over the three-form gauge field A, i.e., after solving the generalized Maxwell equation (4). As a result, the vacuum energy density entering the generalized Einstein equation (5b) is not the original vacuum energy density e(q) from the action (la) but the combination

Pv(q) = e(q) -nq.

(6)

This gravitating vacuum energy density becomes a genuine cosmological constant A = A (q) = pv(q) for a spacetime-independent vacuum variable q.

The field equations (5ab) can now be seen to have a Minkowski-type solution with spacetime-independent fields. For standard global spacetime coordinates, the fields of this constant solution are given by

gaf}{x) = Ца/з , Faf}fS(x) = qo £af}fS ■ ф(х) = Фо ,

(7a) (7b) (7c)

with spacetime-independent parameters /¿o and qo determined by the following two conditions:

d e(q) dq

[ф) ^vq]

= o,

м=мо , q=qo

= 0.

(8a)

(8b)

1 tl=tl 0 , q=q0

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Conditions (8a) and (8b) follow from (5a) and (5b), respectively, for R = Ra@ = Tg-^ = 0 and spacetime-independent qo-

The two conditions (8a), (8b) can be combined into a single equilibrium condition for q0:

An

e(q) -1

d e(q) dq

= 0,

with the derived quantity [26]

Mo =

d e(q) dq

9=90

The spacetime independence of qo implies that of ßo in (10) and, with (5a), guarantees that the generalized Maxwell equation (4) is automatically solved by the Minkowski-type solution (7); see below for a general discussion of this important point. In order for the Minkowski vacuum to be stable, there is the further condition:

a Minkowski-type solution of the field equations. This realization that the g-field allows for a relaxation of the equilibrium condition is the first of the two most important new results of the present Letter.

The Minkowski-type solution of theory (1) is given by the fields (7) with a constant qo parameter that solves ^ (9) and satisfies (11). At this moment, it may be instructive to work out a concrete example. A particular choice for the vacuum energy density function (3) is given by:

e(q) = Abare + (1/2) (Evv)4 sin [q2/(Evv)4 ], (12)

(10) which contains higher-order terms in addition to the standard quadratic term | q2. Needless to say, many other functions e(q) can be chose

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