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The understanding of the gravitational role of vacuum fluctuations is in general a difficult problem, since their energy density usually depends on the renormalization method used and on an adequate definition of the vacuum state in the curved background. In the case of free massless fields in de Sitter spacetime, the renormalized vacuum density is \(\varLambda \approx H^4\) [1–4], which in a low-energy universe leads to a too tiny cosmological term.

In the case we consider the vacuum energy of interacting fields, it has been suggested that in a low energy, approximately de Sitter background the vacuum condensate originated from the QCD phase transition leads to \(\varLambda \approx m^3 H\), where \(m \approx 150\) MeV is the energy scale of the transition [5–11]. These results are in fact intuitive. In a de Sitter background the energy per observable degree of freedom is given by the temperature of the horizon, \(E \approx H\). For a massless free field this energy is distributed in a volume \(1/H^{3}\), leading to a density \(\varLambda \approx H^4\), as above. For a strongly interacting field in a low energy space-time, on the other hand, the occupied volume is \(1/\mathrm{m}^3\), owing to confinement, and the expected density is \(\varLambda \approx m^3 H\).

Such a late-time variation law for the vacuum term can also be derived as a backreaction of the creation of non-relativistic dark particles in the expanding spacetime [12]. The Boltzmann equation for this process is

$$\begin{aligned} \frac{1}{a^3}\frac{d}{dt}\left( a^3n\right) = \varGamma n, \end{aligned}$$
(1)

where \(n\) is the particle number density and \(\varGamma \) is a constant creation rate. By taking \(\rho _m = nM\), it can also be written as

$$\begin{aligned} \dot{\rho }_m + 3H\rho _m = \varGamma \rho _m, \end{aligned}$$
(2)

where \(M\) is the mass of the created particle. Let us take, in addition to (2), the Friedmann equation

$$\begin{aligned} \rho _m + \varLambda = 3H^2, \end{aligned}$$
(3)

with the vacuum term satisfying the equation of state \(p_{\varLambda } = - \varLambda \). Using (2) and (3) we obtain the conservation equation for the total energy,

$$\begin{aligned} \dot{\rho } + 3H(\rho +p) = 0, \end{aligned}$$
(4)

provided we takeFootnote 1

$$\begin{aligned} \varLambda = 2\varGamma H. \end{aligned}$$
(5)

This is the time-variation law predicted for the vacuum density of the QCD condensate, with \(\varGamma \approx m^3\). Dividing it by \(3H^2\), we obtain

$$\begin{aligned} \varGamma = \frac{3}{2} \left( 1-\varOmega _m \right) H, \end{aligned}$$
(6)

where \(\varOmega _m = 1 - \varOmega _{\varLambda } \equiv \rho _m/(3H^2)\) is the relative matter density (for simplicity, we are considering only the spatially flat case). In the de Sitter limit (\(\varOmega _m = 0\)), we have \(\varGamma = 3H/2\), that is, the creation rate is equal (apart from a numerical factor) to the thermal bath temperature predicted by Gibbons and Hawking in the de Sitter spacetime [13]. It also means that the scale of the future de Sitter horizon is determined, through \(\varGamma \), by the energy scale of the QCD phase transition, the last cosmological transition we have. For the present time we have, from (6) (with \(\varOmega _m \approx 1/3\)), \(H_0 \approx \varGamma \approx m^3\), and hence \(\varLambda \approx m^6\), where \(H_0\) is the current Hubble parameter. The former result is an expression of the Eddington-Dirac large number coincidence [14]. The later—also known as Zeldovich’s relation [15]—gives the correct order of magnitude for \(\varLambda \).

The corresponding cosmological model has a simple analytical solution, which reduces to the CDM model for early times and to a de Sitter universe for \(t \rightarrow \infty \) [16]. It has the same free parameters of the standard model and presents good concordance when tested against type Ia supernovas, baryonic acoustic oscillations, the position of the first peak of CMB and the matter power spectrum [12, 17–21]. Furthermore, the coincidence problem is alleviated, because the matter density contrast is suppressed in the asymptotic future, owing to the matter production [12, 20].

With \(\varLambda = 2\varGamma H\) we obtain, from the Friedmann equations, the solution [16–19]

$$\begin{aligned} \frac{H}{H_0} \approx 1 - \varOmega _{m0} + \varOmega _{m0} (1 + z)^{3/2} \;, \end{aligned}$$
(7)

where here, for simplicity, we have not added radiation. For high redshifts the matter density scales as \(\rho _m(z) = 3H_0^2 \varOmega _{m0}^2 z^3\). The extra factor \(\varOmega _{m0}\)—as compared to the \(\varLambda \)CDM model—is owing to the late-time process of matter production. In order to have nowadays the same amount of matter, we need less matter in the past. Or, in other words, if we have the same amount of matter in the past (say, at the time of matter-radiation equality), this will lead to more matter today. We can also see from (7) that, in the asymptotic limit \(z \rightarrow -1\), the solution tends to the de Sitter solution. Note that, like the \(\varLambda \)CDM model, the above model has only two free parameters, namely \(\varOmega _{m0}\) and \(H_0\). On the other hand, it can not be reduced to the \(\varLambda \)CDM case except for \(z\rightarrow -1\). In this sense, it is falsifiable, that is, it may be ruled out by observations.

The Hubble function (7) can be used to test the model against background observations like SNIa, BAO and the position of the first peak in the CMB spectrum [17–19]. The analysis of the matter power spectrum was performed in [20], where, for simplicity, baryons were not included and the cosmological term was not perturbed. In a subsequent publication a gauge-invariant analysis, explicitly considering the presence of late-time non-adiabatic perturbations, has shown that the vacuum perturbations are indeed negligible, except for scales near the horizon [21].

We show in Table 1 the best-fit results for \(\varOmega _{m0}\) (with \(H_0\) marginalized) with three samples of supernovas: the SDSS and Constitution compilations calibrated with the MLCS2k2 fitter, and the Union2 sample. For the sake of comparison, we also show the best-fit results for the spatially flat \(\varLambda \)CDM model. We should have in mind that the Union2 dataset is calibrated with the Salt2 fitter, which makes use of a fiducial \(\varLambda \)CDM model for including high-\(z\) supernovas in the calibration. Therefore, that sample is not model-independent and, in the case of the standard model, the test should be viewed as rather a test of consistence. From the table we can see that for the model with particle creation the concordance is quite good. For the samples calibrated with the MLCS2k2 fitter it is actually better than in the \(\varLambda \)CDM case. As anticipated above, the present matter density is higher than in the standard case, with \(\varOmega _{m0} \approx 0.45\).

Table 1 \(2\sigma \) limits to \(\varOmega _{m0}\) (SNe \(+\) CMB \(+\) BAO \(+\) LSS)
Full size table

With the concordance values of \(\varOmega _{m0}\) in hand, we can obtain the age parameter of the Universe, as well as the redshift of transition between the decelerated and accelerated phases. They are given, respectively, by [16–19]

$$\begin{aligned} H_0t_0 = \frac{2\ln \varOmega _{m0}}{3(\varOmega _{m0}-1)} , \end{aligned}$$
(8)
$$\begin{aligned} z_T = \left[ 2\left( \frac{1}{\varOmega _{m0}}-1\right) \right] ^{2/3}-1. \end{aligned}$$
(9)

In the case of the SDSS and Constitution samples, this leads to \(H_0 t_0 = 0.97\) and \(z_T = 0.81\), in good agreement with standard predictions and astronomical limits [22]. For \(H_0 \approx 70\) km/(s.Mpc), we have \(t_0 \approx 13.5\) Gyr.

Particle creation is something expected in expanding spacetimes [23]. In spite of the difficulty in deriving the production rate and backreaction in general, this phenomenon may in principle be related with inflation [24] and with the present cosmic acceleration, a possibility already considered in different ways by some authors [25, 26] . We have shown that a constant-rate creation of non-relativistic dark particles at late times leads indeed to a viable concordance model.