Effective cosmological constant from TeV–scale physics
It has been suggested previously that the observed cosmological constant corresponds to the remnant vacuum energy density of dynamical processes taking place at a cosmic age set by the mass scale of ultramassive particles with electroweak interactions. Here, a simple modification of the nondissipative dynamic equations of –theory is presented, which produces a remnant vacuum energy density (effective cosmological constant) of the correct order of magnitude. Combined with the observed value of , a first estimate of the required value of the energy scale ranges from 3 to 9 TeV, depending on the number of species of ultramassive particles and assuming a dissipative coupling constant of order unity. If correct, this estimate implies the existence of new TeV–scale physics beyond the standard model.
pacs:95.36.+x, 12.60.-i, 04.20.Cv, 98.80.Jk
Phys. Rev. D 82, 083006 (2010)
It has been argued by Arkani-Hamed et al. ArkaniHamed-etal2000 () that two fundamental energy scales, the electroweak scale and the gravitational scale , suffice to explain the triple cosmic coincidence puzzle: why are the orders of magnitude of the energy densities of vacuum, matter, and radiation approximately the same in the present Universe? For this explanation to work, the parametric form of the effective cosmological constant (remnant vacuum energy density) must be
If true, formula (1) would be a remarkable explanation of the measured value from observational cosmology, which appears to be of order (setting and referring to, e.g., Refs. Riess-etal1998 (); Perlmutter-etal1998 (); Komatsu2008 () and other references therein).111There have, of course, been many other explanations of the smallness of by an appropriate ratio of energy scales (see, e.g., Sec. X of Ref. RatraPeebles1988 ()), but the relation (1) is special as it carries the ingredients to naturally give the correct orders of magnitude for the present matter and radiation energy densities ArkaniHamed-etal2000 (). However, (1) was not derived convincingly in Ref. ArkaniHamed-etal2000 (), as an unknown adjustment mechanism needed to be invoked.
Subsequently, Volovik and the present author realized KV2009-electroweak () that, in the framework of –theory, there is the possibility of generating a vacuum energy density precisely of the form (1). Here, –theory is a particular approach KV2009-CCP1 () to solving the first cosmological constant problem (CCP1): why is ? The original references on the statics and dynamics of –theory are KV2008-statics () and KV2008-dynamics (), respectively. The second cosmological constant problem (CCP2) is the question addressed here, namely, the actual order of magnitude of , if indeed nonzero.
The positive remnant vacuum energy density obtained in Ref. KV2009-electroweak () relied crucially on Eq. (4.1) of that article. That particular equation was taken to describe the quantum-dissipative effects of the vacuum energy density, but was, in the end, purely hypothetical and disconnected from the previous –theory discussion.
The question arises if it is at all possible to modify the previous –theory equations KV2008-dynamics () in such a way as to effectively recover the results of Ref. KV2009-electroweak (). The answer is that this is indeed possible, even though the required modifications are quite subtle. The scope of the present article is restricted to finding these appropriate phenomenological equations, rather than establishing the relevant microscopic processes of the underlying theory.
In a way, it can be said that this whole article is about the proportionality constant implicit in (1) and that, within the framework of –theory, the article gives an existence proof for a set of dynamical equations which produces a proportionality constant of order unity.
As the previous discussion makes clear, the present article is a direct follow-up of Ref. KV2009-electroweak (), to which the reader is referred for the original motivation and detailed analysis. This article is, by necessity, rather technical and it may be helpful to give the reader a road map. The material of this article is organized as follows:
- Track 1:
- Track 2:
- Track 3:
The basic idea and main results are presented in Track 1, the dimensionless differential equations and their numerical solution in Track 2, and a more detailed discussion and further refinement in Track 3 (the most realistic calculations are shown in the very last two figures and the very last table of Appendix A). In a first reading, it is possible to follow Track 1 and to add the other Tracks later.
Ii Theoretical framework
This section reviews the main ingredients of the type of theory considered in this article (see Refs. KV2008-statics (); KV2008-dynamics () for details). The particular –theory realization used involves the so-called 4-form field strength DuffNieuwenhuizen1980 (); Aurilia-etal1980 (). Very briefly, the theory is defined over a four-dimensional Lorentzian spacetime manifold and employs a 4-form field strength derived from a 3-form gauge field . The corresponding rank-four tensor can always be written as
with the Levi–Civita tensor density , the determinant of the metric , and the scalar field . The crucial point is that this scalar field is nonfundamental, being built from the metric field and the 3-form gauge field , as will become clear shortly. This 3-form gauge field is considered to be one of the fields which characterize the quantum vacuum at the fundamental microscopic level KV2008-statics ().
The macroscopic effective action of the relevant high-energy fields (here, ) and the low-energy fields (here, and ) is taken to be of the following form KV2008-dynamics ():
where the effective gravitational coupling parameter is allowed to depend on , is the Ricci curvature scalar obtained from the metric , denotes the standard covariant derivative, and the square bracket around spacetime indices stands for complete antisymmetrization. The energy density is assumed to be a generic function of , that is, a function different from the simple quadratic corresponding to a Maxwell-type theory DuffNieuwenhuizen1980 (); Aurilia-etal1980 (). The field in (3a) stands for a generic low-energy matter field with a scalar Lagrange density , which, for simplicity, is assumed to be without explicit –field dependence (the dependence on the metric arises from the covariant derivatives).
Remark that the effective action (3a) corresponds to a Brans–Dicke-type action BransDicke1961 (), but without kinetic term for the (nonfundamental) scalar . For spacetime-independent [that is, ], the effective action (3a) corresponds to the one of standard general relativity with a cosmological constant , where refers to contributions to from the matter Lagrange density .
By taking variations of and in the effective action (3a), generalized Maxwell and Einstein equations can be derived. The generalized Maxwell equation can be solved explicitly and the solution depends on a constant of integration . With the solution of the generalized Maxwell equation, the generalized Einstein equation reduces to the following field equation:
where the combination
plays the role of the gravitating vacuum energy density rather than the single term appearing in the effective action (3a). Furthermore, there is an equation remaining from the particular solution of the generalized Maxwell equation, which reads
The final equations (4)–(6) can be specialized to the case of a spatially flat Friedmann–Robertson–Walker universe. The resulting cosmological equations have been studied in Refs. KV2009-electroweak (); KV2008-dynamics () and it is the aim of the present article to find a modification of them which allows for the generation of a nonvanishing remnant vacuum energy density.
Iii Vacuum dynamics in a flat FRW universe
iii.1 Basic idea
Following Ref. ArkaniHamed-etal2000 (), assume the existence of ultramassive unstable particles (here, called ‘type 1a’) with masses of order and electroweak interactions. Consider a spatially flat Friedmann–Robertson–Walker (FRW) universe and assume the type–1a particles to be effectively in thermal equilibrium at early enough times. Then, the masses of these particles start to affect the Hubble expansion rate when the temperature drops to , corresponding to a cosmic age of order
in terms of the reduced Planck energy,
Note that the definition (7) is motivated by the standard Friedmann equation with and . In the following, the Friedmann equation will be modified, but the order of magnitude (7) remains relevant. Throughout this article, natural units are used with .
Compared to the case of having only ultrarelativistic particles (these lighter particles are called ‘type 1b’ and can be thought to have masses of order ), the change of the expansion rate from the ultramassive type–1a particles can be modeled by a nonzero function . In fact, this function can be written in terms of the standard equation-of-state (EOS) parameter as follows:
which vanishes for ultrarelativistic particles () and equals unity for pressureless nonrelativistic particles (). By taking different unstable type–1 particles () it is possible to obtain an effective EOS function which peaks at . Here, however, a particular form of will simply be assumed.
The main conditions on this assumed EOS function are that it peaks at and is nonzero only in a finite range around the maximum (this last condition is not essential but simplifies the discussion). Specifically, the conditions are taken to be:
with and , having set for the big bang where diverges. The physical picture corresponding to (10) is that ultramassive type–1a particles are dominant at , but, then, decay into lighter type–1b particles which are still ultrarelativistic for not very much larger than . The main goal is to study the effects of this prescribed EOS function , relegating the discussion of a more realistic EOS function to App. A.
A flat FRW universe containing only type–1 particles with a prescribed EOS parameter (9)–(10) has a standard radiation-dominated Hubble expansion rate for and . The expansion rate is changed, , for times between and . The question, now, is what happens if this FRW universe also has a dynamical vacuum-energy-density component.
For the theory outlined in Sec. II, two results were obtained in Ref. KV2009-electroweak (). First, it was shown that there is an exact solution having in the radiation-dominated phase with . Second, it was shown that the changed Hubble expansion from kicks away from zero. Specifically, the following behavior was established KV2009-electroweak ():
which vanishes asymptotically as drops to zero and the standard radiation-dominated expansion of the model universe resumes. At the moment of the kick, , the vacuum energy density (11) is of order , which is negligible compared to the matter energy density . The vacuum energy density , therefore, just responds to (is being kicked by) the Hubble expansion and does not affect the expansion substantially.
The result (11) has been obtained from the simplest time-reversible (nondissipative) version of –theory, with field equations given by (4)–(6). It has been argued that quantum-dissipative effects (e.g., because of particle production in an expanding universe ZeldovichStarobinsky1977 (); BirrellDavies1982 ()) may result in a freezing of the previous result (11) to a constant nonzero value.
As explained in Sec. I, the aim of this article is to find a suitable modification of the “classical” –theory equations, which produces a finite remnant vacuum energy density. In the approach followed here, there are three changes:
The matter energy-conservation equation is modified to include appropriate particle-production effects operating at a cosmic age .
The reduced Maxwell equation is modified, so as to match the standard Einstein equation of an FRW universe with a nonzero effective cosmological constant at later times.
Different particle species are considered with ultramassive type–1a particles first decaying into lighter type–1b particles, which, in turn, decay into massless type–2 particles.
The first two modifications are essential for the generation of a nonvanishing remnant vacuum energy density. (The first modification has already been discussed in general terms in Sec. IV of Ref. KV2009-electroweak ()). The third modification allows for a possibly more realistic scenario, with type–1 particles corresponding to new TeV-scale physics and type–2 particles corresponding to the standard model of elementary particle physics (see also App. A).
The particle-production effects of point 1 above will be controlled by an effective coupling constant and a particular type of dissipation function . The reason for calling a “dissipation” function will become clear in Sec. III.5. The main conditions on this function are as follows:
with a particular time that is of order for and approaches infinity for .
The coupling constant and the corresponding function are purely phenomenological. The vanishing of for large enough times will be seen to have two effects: first, to freeze the “classical” value (11) and, second, to switch to a standard FRW expansion with relativistic matter and a tiny value (1) for the remnant vacuum energy density. Further discussion of and will appear in the next subsection, after the differential equations have been presented.
iii.2 Modified ODEs
For a spatially flat FRW universe and the 4-form realization of –theory with a variable gravitational coupling parameter , the cosmological differential equations have been derived in Ref. KV2008-dynamics () and were already mentioned in Sec. II. The basic idea of the proposed modification of these ordinary differential equations (ODEs) has been discussed in the previous subsection. Specifically, the modified ODEs are given by:
where only the arguments of the functions and have been shown explicitly. The four equations in (13), going from the top to the bottom, can be recognized as modified versions of the reduced Maxwell equation (6), the two matter energy-conservation equations, and the Friedmann equation [the standard Friedmann equation is recovered for ]. For and , the ODEs (13) correspond to Eqs. (4.12abc) of Ref. KV2008-dynamics (), supplemented by an equation for the adiabatic evolution of .
The modified ODEs for have a dependence on an external time scale, here taken to be from (7). The dependence enters implicitly through the EOS function and the dissipation function discussed in the previous subsection and explicitly through the matter energy-exchange terms proportional to . Note that the dimension of the –field in Eqs. (13) is irrelevant, which concords with the fact that this –field may be realized in different ways KV2008-statics ().
The particular modification (13) of the cosmological ODEs from classical –theory has two main ingredients: first, the function with characteristics (12) and, second, the presence of a finite coupling constant ,
which enters directly on the right-hand side of (13b) and indirectly via condition (12c) for the dissipation function . A finite value for the remnant vacuum energy density from the dynamical ODEs (13) requires both and , as will be explained in Sec. III.5.
The rather simple structure of (13), combined with conditions (10) and (12), will be seen to allow for the generation of a nonzero remnant vacuum energy density.222A somewhat more general modification of the ODEs has in the last term on the right-hand side of (13a) replaced by , but, for the case considered in this section and Sec. IV, the results are essentially unchanged, because for according to (10). A detailed discussion of the modified ODEs is postponed until Sec. III.5, after these equations have been established in dimensionless form.
Before embarking on this technical enterprise, it may be useful to recapitulate the basic assumptions. The first assumption is the existence of a particular type of vacuum variable , namely, a variable which corresponds to a conserved relativistic quantity in flat Minkowski spacetime. Such a variable provides a possible solution of the main cosmological constant problem (CCP1) by explaining why is naturally zero in the equilibrium state.333The –theory approach to CCP1 provides only a possible solution, because it is not known for sure that the underlying microscopic theory does contain an appropriate –type field. In addition, there remain other equally fundamental (perhaps related) questions, such as the nature of gravity and the origin of spacetime. The goal of the present article is relatively modest: to explore, in the framework of –theory, a possible connection between the observed value of the effective cosmological constant and new TeV–scale physics. This vacuum variable is taken to have an effective action of the form of (3), where, in particular, the gravitational coupling constant may carry a dependence on .
The second assumption is that the field equations from the effective action (3), specialized to a spatially flat FRW universe, are modified by the introduction of terms involving the coupling constant . The crucial term is the first one on the right-hand side of (13b), whose physical motivation is that it reproduces the dissipative behavior suggested in Ref. KV2009-electroweak () (this behavior is analogous to that of bulk viscosity in compressible material fluids LandauLifshitz-Fluid-mechanics ()). The coupling constant and the corresponding function are purely phenomenological. As mentioned in Sec. I, ultimately and (or appropriate generalizations) need to be derived from the underlying microscopic theory, but that task lies outside the scope of the present article.
Clearly, the first assumption is better motivated than the second. But the second assumption may (or may not) gain in credibility depending on the success (or not) of producing a reasonable remnant vacuum energy density and predicting new TeV–scale physics.
iii.3 Ansätze and dimensionless variables
These Ansätze imply the following equilibrium value for the –field of theory (3):
where is the energy scale from (8). In this article, is considered to be realized by a 4-form field strength with mass dimension 2. With a proportionality constant in (15b) of order unity, the natural scale of the vacuum variable is then of order . However, the energy scale of in the cosmological ODEs (13) is, in principle, arbitrary, as noted already in the second paragraph of Sec. III.2.
Next, recall the time scale defined by (7), which corresponds to the age of the Universe at a temperature of order . For later use, also define the following number characterizing the hierarchy of energy densities:
which is of order for . For such a large value of , the cosmic time considered in this article is large compared to the Planck time, . In addition, the relation can be seen to hold, which will be used later for the derivation of the dimensionless ODEs.
With and , the following dimensionless variables can be defined for the cosmic time, the Hubble expansion rate, the energy densities, and the shift away from equilibrium:
At this moment, it is appropriate to give explicit examples for the EOS function and the dissipation function discussed in Sec. III.1. With central value and total width , define the auxiliary variable and take the EOS function to be given by
In the main part of this article, is set to and the dynamic strength of the kick is controlled by the value of the initial energy-density ratio (see App. A for a more realistic EOS function).
Turning to the dissipation function , introduce the basic time scale , define
|and take the function to be given by|
The two functions used will also be shown in the plots of the numerical results later on.
iii.4 Dimensionless ODEs
Take, now, the Ansätze (15a)–(15b) and assume small deviations of away from the equilibrium value , i.e., . Then, the ODEs (13) reduce to the following four equations for the four dimensionless variables , , , and from (18):
where the overdot stands for differentiation with respect to . Henceforth, the functions and are considered to be given by the explicit expressions (19) and (20). For the numerical calculation, the factor on the right-hand side of (21b) is to be replaced by the appropriate expression for obtained from (21d). Recall, furthermore, that the dimensionless vacuum energy density is given by and that, according to (18b), the dimensionless matter energy densities and include an extra numerical factor compared to .
For later use, also the ODEs for the special case are needed. From (21), the following system of equations can be derived for , with three ODEs:
and a single algebraic equation:
The derivation of (22b) proceeds in three steps: first, take the time derivative of (23); second, use (22a) to eliminate in the resulting expression for ; third, use the final expression for in the sum of the two Eqs. (21b) and (21c) to get (22b). All in all, the equations consist of three ODEs (22a)–(22c) for three variables , , and , with the energy density following from the Friedmann Eq. (23). These ODEs will be used in Sec. IV.2 to get the value of the remnant vacuum energy density for .
Purely mathematically, there is another special case to consider for the ODEs (21) as they stand, namely, the case . From (21a) and (21d), together with the proper boundary condition , the solution can be seen to have , which implies . But, as said, this solution is not directly relevant for the physical situation considered.
iii.5 Additional remarks
which corresponds to Eqs. (3.1a) and (3.4a) of Ref. KV2009-electroweak () apart from a trivial rescaling. With dropping to zero rapidly for large enough , there is no sizable remnant vacuum energy density, at least, according to the unmodified ODEs given by (21) for and .
For nonzero , however, the same approximations in the ODEs (21) give the following dissipation-type equation at :
whose derivation parallels the one of (22b) in the previous subsection. Equation (25a) with boundary condition is the analogue of the crucial relation (4.1) of Ref. KV2009-electroweak () that allows for a positive remnant vacuum energy density as discussed in Sec. IV of that article. Section IV of Ref. KV2009-electroweak () contains, in fact, the analytic solution of (25a) for given functions , , and . For completeness, the dimensionful quantity corresponding to is given by .
The dissipative ODE (25) and its analytic solution KV2009-electroweak () make clear that a finite remnant vacuum energy density with requires both and . Indeed, for the case (or ), the solution follows which drops to zero rapidly for large times and, for the case (or ), the solution simply remains at the initial value, .
Second, it appears essential that the ODEs (13) and (21) are singular, with the coefficients of the first terms on the right-hand sides of (13b) and (21b) diverging for , because of the condition (12b). In fact, the divergence of the coefficient for above forces to be strictly constant. It could very well be that the exact vanishing of for in the cosmological context traces back to the existence of an energy threshold in the relevant particle reaction process. As noted in Ref. KV2009-electroweak (), the energies involved are tiny (of the order of meV), so that only sufficiently light neutrinos and gravitons can be expected to play a role. The first term on the right-hand side of (13b), as it stands, is not simply proportional to as for the well-known Zeldovich–Starobinsky result ZeldovichStarobinsky1977 (), but does involve via its time derivative, as follows by use of (13a).
Third, the discussion of the two previous remarks suggests that the value of the remnant vacuum energy density can be at most of the order of the maximum possible “classical” value (i.e., the peak from the nondissipative equations). The idea is that, in general, dissipation leads to reduction of the produced energy rather than enhancement. Specifically, the conjectured inequality is
which is based on the analytic result (24b) with on the right-hand side replaced by , as suggested by (22b). It remains to sharpen the approximate upper bound (26) and to determine the corresponding conditions.
Fourth, having a constant nonzero value of does not automatically allow for a standard de-Sitter universe, as the original ODE (21a) [with ] and the ODE (21d) are inconsistent for , , and . However, the modified Eq. (21a) [with for ] has been designed to match the corresponding Einstein equation of a standard flat FRW model with ultrarelativistic matter and constant vacuum energy density, which asymptotically approaches a de-Sitter universe.444The assumption, here, is that other contributions to the vacuum energy density generated at times later than would be self-adjusted away by appropriate –type fields KV2008-statics (). The prime example is the quantum-chromodynamics (QCD) vacuum energy density of order , which is expected to appear during the cosmological QCD transition at . This huge contribution to the vacuum energy density has been shown to self-adjust to zero KV2009-gluoncondcosm (); Klinkhamer2009 (), as long as there is no term proportional to contributing to . If there is such a nonanalytic term (cf. the discussion in Ref. UrbanZhitnitsky2009 ()), then the final value of could be a combination of electroweak and QCD effects. However, the experimental signatures of the electroweak vacuum energy density (effectively, a CDM model, as explained in the next footnote) and the QCD vacuum energy density (an modified-gravity model) are different, in particular as regards the effective EOS parameter discussed in Ref. Klinkhamer2009 (). But, for the moment, there is no definitive proof that the required nonanalytic term occurs in four-dimensional QCD. This particular modification also makes clear that there must be more than just energy exchange between the vacuum and matter sectors. Rather, there must be a type of modulated interaction between the vacuum field and the nonstandard gravitational field, which can be seen as follows. Multiply (13a) by to get a modified FRW–Einstein equation,
where the nonstandard term is switched off for large enough cosmic times by the factor going to zero.
Fifth, the ODEs (21), for given functions and and fixed coupling constants and , contain one last free parameter, the hierarchy parameter defined by (17). Heuristically, it is to be expected that the precise value of does not affect the resulting value . But, as fixes the ratio at , it does affect the later (standard) evolution of the model universe and, in fact, determines ArkaniHamed-etal2000 () the cosmic time at which the matter energy density drops below that of the constant vacuum energy density, . More precisely, the onset of acceleration [having defined in terms of the scale factor ] occurs at the energy-density ratio , with for relativistic matter ( and ) or for nonrelativistic matter ( and ). The model considered in the present article has , but can easily be adapted to give the value which is more realistic.555The adapted model contains an additional EOS function , which is a smoothed step function running from to as the cosmic time increases and which has a half-way time where . This matter-radiation-equality time has a parametric form , with the electromagnetic fine-structure constant (see Ref. ArkaniHamed-etal2000 () for further details). The adapted model then has an acceleration phase for and corresponds, for , to a particular CDM model Komatsu2008 (); Mukhanov2005 (). As noted in Ref. KV2009-electroweak (), this EOS function can also be expected to perturb the vacuum energy density, but the magnitude involved is tiny compared to the one from the electroweak scale because .
Iv Numerical solution
iv.1 Numerical results for
The mathematical parameter entering the ODEs (21) is, first, considered to have the moderately large value of 100. The boundary conditions are taken from the epoch before the electroweak kick, when there was a standard radiation-dominated flat FRW universe. With the onset of the electroweak kick given by from (10b) and (19), the following boundary conditions on , , and hold at a time :
This leaves only the initial ratio to be determined, which is, for the moment, taken to be (other initial ratios will be discussed shortly).
The corresponding numerical solutions of the ODEs (21) are shown in Figs. 1–3. Figure 1 illustrates the fact that the standard (nondissipative) dynamic equations for and do not produce a constant positive remnant vacuum energy density from the electroweak kick [the oscillatory effects in are suppressed for larger values of the hierarchy parameter ; see the first figure called in Sec. IV.2]. However, as shown by Fig. 2, the modified (dissipative) dynamic equations for do produce a sizable remnant vacuum energy density.
The subsequent evolution of Fig. 2 is shown in Fig. 3. The content of this model universe for is given by a constant vacuum energy density (effective cosmological constant) and two species of matter, with massive type–1 particles playing a role for the generation of the vacuum energy density during the electroweak epoch () and ultimately decaying into massless type–2 particles.
Similar results are obtained for initial ratios . Table 1 gives the function values at cosmic time , in particular, the values for which, by construction, stay constant for later times . Remark that even for a relatively mild kick with initial ratio , the generated is still of order .
Returning to the boundary condition , Figs. 2–3 are seen to give a value . By changing the model parameters and the model functions somewhat it is possible to get values in the range of to . But it appears impossible to get a remnant much larger than unity, which agrees with the conjectured upper bound (26).
iv.2 Numerical results for
The parameter has been defined in physical terms by (17) and its mathematical role for the solution of the ODEs (21) has already been discussed in the last paragraph of Sec. III.5. Here, numerical results are presented for large values of this parameter, ranging from to .
Numerical results for the standard nondissipative () dynamic equations at are given in Fig. 4, which show reduced oscillatory effects of compared to Fig. 1 and recover the smooth behavior of (24b). [Recall that the analytic approximation (24b) was derived for a negligible type–2 matter energy density and a better approximation has on the right-hand side replaced by , as used already in (26).] Further numerical results for and confirm the expectation from Sec. III.5 that the generation of the remnant vacuum energy density at is qualitatively the same as for (compare Fig. 5 with Fig. 2) and that the main effect of a larger value of is that of pushing the onset of the vacuum-dominated expansion to larger values of , with (compare Fig. 6 with Fig. 3). It is also instructive to contrast the behavior of the vacuum variable in Fig. 4 and that of Fig. 5, where the latter figure displays the “time-lag effect” because of the finite dissipative coupling constant (cf. the heuristic discussion of the paragraph starting a few lines under Eq. (4.5) in Ref. KV2009-electroweak ()).
As the model evolution for is perfectly standard (described by an FRW universe with ponderable matter and an effective cosmological constant), the crucial segment of the numerical solution is over the interval . The numerical data for the function values are given in Table 2, where the values refer to the solution of the equations (22) derived in Sec. III.4. The functions from Table 2 are observed to converge for . The convergence of the vacuum energy density results is also shown by Fig. 7. With values for parameters both below and above the “realistic” number , the following estimate is obtained by interpolation:
for the model parameters and boundary conditions mentioned in the caption of Table 2.
The function values at calculated from the ODEs (22) are given in Table 3 for a wide range of values of the initial ratio at . These function values can be expected to approximate the physical () values with an accuracy of one per mill or better, at least, for initial ratios of order unity.
As far as the dimensionless remnant vacuum energy density is concerned, the values quoted in Table 3 constitute the complete solution of the problem where the initial ratio controls the relative strength of the kick at , assuming the validity of the phenomenological ODEs (13) for coupling constant and taking model functions and from (19) and (20), respectively. The dimensionful remnant vacuum energy density requires knowledge of the absolute energy scale used in the rescaling of the variables, as will be discussed further in the next section.
v.1 General case
with a number obtained by numerically solving ODEs of the type of (13). Equating the theoretical value (30) with the experimental value from observational cosmology Riess-etal1998 (); Perlmutter-etal1998 (); Komatsu2008 (), the following estimate for the required energy scale is obtained:
For the moment, the numerical value in (31) is purely for illustrative purpose.
Clearly, the calculation of the present paper relies on many assumptions, but it appears that values of the order of unity for the dimensionless energy density are quite reasonable. The value corresponds to , according to (31). On the other hand, values appear unlikely, at least, in the present framework [see the discussion in Sec. III.5 leading up to (26)]. Note that an value of order would be required in (31) to bring down to the order of magnitude of standard-model particle masses, .
Taking for granted, the correct reading of (31) is that of a lower bound,
since, without further input, the value of can be made arbitrarily small (for example, by taking a sufficiently small value for the initial energy density in Table 3). Indeed, the main uncertainty (apart from the unknown value) is the dynamic importance at of the nonrelativistic () type–1 particles compared to that of the relativistic type–2 particles.
v.2 Special case
In order to get further predictions, the following three assumptions can be made. First, assume the type–1 and type–2 particles to have been ultrarelativistic and in thermal equilibrium for , so that their energy densities are given by
|with bosons () and fermions () of particle type contributing as follows:|
|in terms of the numbers of degrees of freedom of the particles ( for the photon). Then, the relevant energy-density ratio before the kick starts is simply given by the ratio of the respective effective numbers of degrees of freedom, .|
|Second, assume the type–1 particles to have approximately the same mass and a mass scale|
|As discussed in Sec. III.1, these type–1 particles can be thought to consist of a mix of different unstable particles. What matters here, though, is their average thermodynamic properties as given by the prescribed EOS function from (19). See App. A for a realistic setup.|
|Third, assume the massless type–2 particles to correspond to those of the standard model (), so that|
See, e.g., Ref. Amsler-etal2008 () for the count of the degrees of freedom in the standard model.
With these assumptions, there are only two unknowns: the numerical value of the energy scale and the effective number of type–1 degrees of freedom. The first quantity, , sets, according to (7), the physical time when the ‘kick’ of the vacuum energy density occurs (the kick mechanism KV2009-electroweak () relies on the change of the Hubble expansion rate by type–1 mass effects). The second quantity, , controls the initial energy-density ratio:
where the dimensionless cosmic time is taken before the kick starts and denotes the dimensionless energy density according to (18).
For a substantial number of these ultramassive type–1 particles (possibly corresponding to partners of the standard-model particles from broken supersymmetry WessZumino1974 (); FayetFerrara1976 ()), the initial energy-density ratio is given by and the dimensionless remnant vacuum energy density is found to be of order [see Table 3]. This particular value requires, according to (31), an value of order , in order to reproduce the experimental value of the cosmological constant. A similar value is obtained in App. A if the prescribed (artificial) EOS function (19) is replaced by a physically-motivated EOS function. Only for (corresponding to a single ultramassive real scalar) does the remnant vacuum energy density drop to such a low value, [Table 3], that the required energy scale becomes significantly larger, .
Hence, if all particles have initially been in thermal equilibrium and if the type–1 particles with mass scale have an effective number of degrees of freedom , the required energy scale from (31) lies in the following range:
assuming a dissipative coupling constant of order unity. The trend is, as expected from (30), that a smaller number requires a larger energy scale . Table 4 presents the required energy scales for selected values of , with the understanding that the quoted numbers for are only indicative because of the many assumptions made along the way (some of which may be more reasonable than others). See App. A for further discussion of some of the systematic uncertainties involved (its very last table complements Table 4 of this section).
If the observed “cosmological constant” results from dynamics at cosmic temperatures of order , then some set of differential equations must be relevant. In the framework of –theory, a particular set of differential equations has been proposed. It appears to be impossible to have very much simpler differential equations which achieve the same result and the phenomenological equations used here can be expected to carry some of the essential ingredients. If so, the estimates from Table 4 suggest the need for new physics with particle masses at the TeV–scale.
Particle-collider experiments are called upon to confirm or exclude the existence of these TeV–scale particles and, if confirmed, to determine their characteristics. Knowing the characteristics of the new TeV-scale particles (assuming their detection), the main task for theorists would be to derive the relevant particle-production effects contained in the simple phenomenological equations considered here or to find the appropriate generalizations of these equations.
It is a pleasure to thank G.E. Volovik for numerous discussions on vacuum energy over the years and the referee for helpful comments on earlier versions of this article.
Appendix A Dynamic kick
The goal of this appendix is to present a calculation for the generation of the remnant vacuum energy density by a more or less realistic kick from dynamically generated ultramassive and unstable type–1 particles. The description of this dynamic kick is rather involved, but the final ODEs are similar to those of the main text for a prescribed kick, with the crucial role again being played by the phenomenological dissipation function (the other functions entering the ODEs will now be given different notations, for clarity). The heuristics of these ODEs will be discussed in the last paragraph of Sec. A.4.
a.2 Mass spectrum and EOS functions
The massless type–2 particles from the model introduced in Sec. III.1 are considered to correspond to those of the standard model (mass scale ) and the round number will be used. The ultramassive type–1 particles are considered to arise from broken supersymmetry WessZumino1974 (); FayetFerrara1976 (), so that , and they can be taken to be bosons (the standard-model particles being mostly fermions Amsler-etal2008 ()). Just as discussed in Sec. V.2, the type–1 and type–2 particles are assumed to have been in thermal equilibrium before the generation of the vacuum energy density starts.
A general type–1 mass spectrum is given by the effective numbers of particles with dimensionless masses , for which the following two constraints hold:
where the last constraint ensures that the average dimensionful mass equals .
For simplicity, consider two cases: case A with two different mass values and case B with a single mass value, by definition equal to . (The generalization to a general type–1 mass spectrum will be obvious.) The specific numbers are chosen as follows:
The case–B partition of 100 is arbitrary, at least, for the dynamic ODEs considered here. It will be seen, later on, that case A and case B give more or less the same remnant vacuum energy density, which even holds for the extreme version of case–A having 50 particles with and 50 with .
Next, take, instead of the prescribed EOS function (19) used in the main text, the exact EOS function or, at least, a controlled approximation of it. In fact, the following rational function of the variable will be employed for type–1 subspecies :