Deflation of the cosmological constant associated with inflation and dark energy
In order to solve the fine-tuning problem of the cosmological constant, we propose a simple model with the vacuum energy non-minimally coupled to the inflaton field. In this model, the vacuum energy decays to the inflaton during pre-inflation and inflation eras, so that the cosmological constant effectively deflates from the Planck mass scale to a much smaller one after inflation and plays the role of dark energy in the late-time of the universe. We show that our deflationary scenario is applicable to arbitrary slow-roll inflation models. We also take two specific inflation potentials to illustrate our results.
It is well-known that there exist two accelerating expansions in our universe. One occurs at the very early time of the universe, solving particularly the flatness and horizon problems, called “Inflation” Starobinsky:1980te (); Guth:1980zm (); Starobinsky:1982ee (); Linde:1983gd (), and the other is at the late-time, indicated by the type-Ia supernovae observations Riess:1998cb (); Perlmutter:1998np (), called “Dark Energy.” The former epoch can be realized by introducing the inflaton field with a slow-roll potential. For the latter, various theoretical models have been proposed to achieve the late-time acceleration universe Copeland:2006wr (), while the simplest one is to keep the cosmological constant, , in the gravitational theory, referred to as the CDM model. However, if is originated from the vacuum energy, it is associated with the Planck mass as predicted in particle physics, which is about orders of magnitude larger than the current measured value. This “fine-tuning” or “hierarchy” problem in fact was known even before the discovery of dark energy WBook ().
There have been many attempts to solve the hierarchy problem Review1 (). One of the popular ways is the running model, in which the vacuum energy decays to matter Ozer:1985ws (); Carvalho:1991ut (); Lima:1994gi (); Lima:1995ea (); Overduin:1998zv (); Carneiro:2004iz (); Shapiro:2004ch (); Bauer:2005rpa (); Alcaniz:2005dg (); Barrow:2006hia (); Shapiro:2009dh () or a quintessence field Costa:2007sq () in the evolution of the universe, and its observational tests have been extensively investigated in the literature EspanaBonet:2003vk (); Borges:2007bh (); Borges:2008ii (); Tamayo:2015qla (); Sola:2016vis (). In this study, we concentrate on a simple model with the vacuum energy non-minimally coupled to the inflaton field, in which the coupling may arise from the conformal transformation, i.e., transforming the Brans-Dicke theory from the Jordan frame to the Einstein one Peebles:1999fz (); Copeland:2000hn (); Sahni:2001qp (); Sami:2004xk (); Hossain:2014zma (); Geng:2015haa (). It is interesting to mention that the inflaton coupled to the vacuum energy can be used to realize inflation with a small coupling constant Nishioka:1992sg ().
In our model, the vacuum energy begins with the Planck mass scale and deflates by decaying to the inflaton in the pre-inflationary and inflationary epochs. The non-minimal coupling between the inflaton and vacuum plays the role of heating up the inflaton and triggers inflation, whereas inflation itself is driven by another slow-roll potential. After the reheating era, the inflaton decays to matter and decouples to the vacuum energy. As a result, the residual vacuum energy density is much smaller than the matter one after inflation. At the late-time of the universe, the vacuum energy is dominated again, known as dark energy. In this scenario, the energy difference between the Planck mass and current cosmological constant scale is determined by the inflaton potential and the coupling constant between the vacuum energy and inflaton field. We will show that the allowed range of the coupling constant is insensitive to the choice of the potentials so that our deflation scenario for is quite general.
This paper is organized as follows: In Sec. II, we introduce the model with the vacuum energy non-minimally coupled to the inflaton field. In Sec. III, we calculate the analytical solution and estimate the range of the coupling constant. In Sec. IV, we use two specific potentials to check our analytical results. We present our conclusions in Sec. V.
Ii Non-minimally coupled vacuum energy and inflaton
We start from the Brans-Dicke action with the vacuum energy Hossain:2014zma (),
where is the Planck mass, is the metric in the Jordan frame, is the Brans-Dicke parameter, and is the Lagrangian density of the vacuum energy. By using the conformal transformation
the Brans-Dicke action is transformed from the Jordan frame into the Einstein one, given by
where is the Einstein frame metric, is the inflaton field, and is a slow-roll inflaton potential.
By varying the action (4) with respect to the metric and specializing to the FLRW case with , we obtain,
where and are the energy density and pressure of the inflaton, defined by
while and are the energy density and pressure of the vacuum, respectively. Note that the vacuum energy equation of state (EoS) satisfies the relation, . If we take the conformal transformation coefficient in Eq. (4) to be,
with being the model parameter to be determined, the inflaton field and the continuity equations for the vacuum energy can be derived as,
respectively. Thus, the vacuum energy can be solved as the function of from Eq. (10),
where is the vacuum energy density after the Big Bang. Consequently, the inflaton field equation in Eq. (9) becomes
where the combined potential is given by
Iii Analytical estimations with generic potential
In Eq. (8), we consider , so that the vacuum energy decays to the inflaton field in the pre-inflation and inflation eras. After the reheating epoch, the inflaton decays to standard model particles, and the vacuum energy decouples to the inflaton field. As result, the original cosmological constant at the Planck mass scale is deflated to the current small measured value of . Explicitly, we have
where is the minimum of the inflaton potential and is the current dark energy density. From Eq. (14), we get
In Fig. 1, we illustrate the potential as a function of the inflaton , where the solid, dashed and dotted lines correspond to the combined potential , the effective potential and the slow-roll potential in Eq. (13), respectively. The evolution history can be divided into the following four stages.
(i) the epoch of the vacuum energy decay
At the first stage, the vacuum energy decays to the inflaton field with and ends up at with (see also Fig. 1). In this period, the evolution of is dominated by , indicating that the inflaton is heating up by the vacuum energy decay. The growth of the inflaton is given by
which is much smaller than in Eq. (15). We note that since and by the end of this stage, the energy flow from the vacuum energy to the inflaton slightly influences the evolution of after this stage.
(ii) the fast-roll pre-inflationary epoch.
In this stage, the universe is undergoing the decelerating pre-inflationary era and dominated by the “hot” inflaton, corresponding to . The energy density follows the continuity equation and is diluted by the expansion of the universe with . This stage is terminated when the slow-roll inflation occurs at , i.e., , leading to
where is the potential energy during inflation (see also Fig. 1). If we consider that the value of at the beginning and end of inflation are of the same order of magnitude, the scale of the potential energy is , where is the Hubble parameter at the inflationary-era (end) of inflation with given in Ref. Liddle:1993ch ().
Note that the equation of state of the inflaton is given by
in this stage. The inflaton energy decreases from the Planck scale to the inflation scale at the end of (ii), resulting in that
which gives the e-folding during this stage to be . Combining with Eq. (18), we find that the stage (ii) is terminated at . We note that the inflationary energy scale is model-dependent, but the choice is insensitive to . For example, if GeV and , the growths of are of the same order of magnitude.
(iii) the slow-roll inflationary epoch.
In this epoch, inflation is triggered and the evolution of the universe depends on the inflation model. This stage ends up with the increasing of the e-folding and
where is the slow-roll parameter. The growth of the inflaton is extremely model-dependent in this period. However, it is still possible to estimate its value by taking the slow-roll condition, i.e.
In addition, the tensor-to-scalar ratio is calculated by . Clearly, it is reasonable for us to have the estimation of
with in this stage. As a result, the growth of the inflaton is by taking during inflation. We note that the allowed , depending on the specific inflation model, can be bigger, but the order of magnitude should be the same. By considering various reasonable parameters, we take
(iv) the reheating epoch
Here, is also model dependent. We assume that the potential can be expressed as around the potential minimum. If we take as the condition at the end of inflation, we can deduce that .
Iv Numerical results with specific potentials
To check our estimations in Sec. III, we present the numerical evaluations with two specific slow-roll potentials:
In Fig. 2, we demonstrate the evolutions of (a) the slow-roll parameter and (b) as functions of with the potential in Eq. (28) and . From the figure, we can see the evolution behaviors of from the stage (i) to (iv). Note that we have taken that inflation occurs at and ends up at , while the mass hierarchies are and . In Fig. 3, we illustrate the evolutions of and as functions of the e-folding . From the plots, we can see that the decaying vacuum energy heats up the inflaton in the stage (i) with . The inflaton energy approaches the Planck scale at the beginning of the stage (ii) and is diluted by the expansion of the universe as with and . In addition, we observe that keeps to be constant in the inflationary era in the stage (iii) until and evolves to the reheating epoch in the stage (iv). On the other hand, decreases in the stages (i - iii), whereas it is nearly constant in the stage (iv), implying that the detail of the reheating is insensitive to the final value of .
In Fig. 4, we use (4.09, 4.20, 4.00), which are represented by the solid, dashed and dotted lines, respectively, and , corresponding to the inflation energy at GeV. In Fig. 4a, we plot the evolution in terms of the e-folding , and we can observe that inflation occurs at and ends up at , which fit our predictions of and . Fig. 4b shows the e-folding versus the normalized physical time , where is the Planck time, and the corresponding e-foldings are , and , respectively. The figure indicates that the e-folding is one-to-one correspondence to the coupling constant .
We now examine the large field inflation potential Linde:1983gd () in Eq. (29) with GeV. In Fig. 5, we plot the evolutions of (a) and (b) as functions of and , respectively. In the figure, we choose , denoted by solid, dashed and dotted lines, respectively. From Fig. 5a, we find that inflation happens and ends up at with , resulting in , which is the same order as our expectation. As shown in Fig. 5b, the corresponding e-foldings are , and , respectively, which also illustrate the one-to-one correspondence to the coupling constant .
We have proposed a deflationary cosmological constant model to understand the hierarchy problem, in which the vacuum energy non-minimally couples to the inflaton field. In this model, the energy difference between the Planck mass and the current scale of the cosmological constant is determined by the non-minimal coupling constant and the inflaton at the minimum of the potential . These two parameters are no longer to be unreasonable huge (or small), so that the hierarchy problem can be resolved. Explicitly, we have estimated that the allowed range of and are around and , which depend mildly on the inflation models. Our results have also been supported by the numerical calculations of the two popular slow-roll potentials.
This work was partially supported by National Center for Theoretical Sciences, National Science Council (NSC-101-2112-M-007-006-MY3), MoST (MoST-104-2112-M-007-003-MY3) and National Tsing Hua University (104N2724E1).
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