Now we begin with the following three stories:
Story 1
The first dream that I believe in is the holographic dark energy model [1–5] based on the holographic principle [6], so let us start from here. In my original paper [1], when the scale factor is small enough, the ratio between the dark energy density and the critical density is
$\begin{eqnarray}{{\rm{\Omega }}}_{{D}}\sim {a}^{-2},\end{eqnarray}$
when the inflation starts, at time 1, and when the inflation stops, at time 2.According to equation (1.1 ),
$\begin{eqnarray}\displaystyle \frac{{{\rm{\Omega }}}_{{D}_{1}}}{{{\rm{\Omega }}}_{{D}_{2}}}={\left(\displaystyle \frac{{a}_{2}}{{a}_{1}}\right)}^{2}.\end{eqnarray}$
Therefore,
$\begin{eqnarray}\displaystyle \frac{{{\rm{\Omega }}}_{{D}_{f}}}{{{\rm{\Omega }}}_{{D}_{i}}}={\left(\displaystyle \frac{{a}_{i}}{{a}_{f}}\right)}^{2}={{\rm{e}}}^{-2N}.\end{eqnarray}$
Let ${{\rm{\Lambda }}}_{D}$ be the infrared energy cutoff corresponding to the dark energy, then ${{\rm{\Lambda }}}_{D}={10}^{12+n}\,$GeV $=\,{10}^{n-7}{M}_{p}$ = ${10}^{n-7},$ where we set the Planck scale to be 1. For simplicity, we ignore the order one factors. Then we infer the following: $\begin{eqnarray}2N=4\left({n}_{{D}_{i}}-{n}_{{D}_{f}}\right).\end{eqnarray}$
During inflation, the driving energy density is $\rho ={\left({{\rm{\Lambda }}}_{{\rm{\inf }}}\right)}^{4},$ where ${{\rm{\Lambda }}}_{inf}$ is the UV cutoff of the inflation, which can be expressed as ${10}^{{n}_{{\rm{\inf }}}-7},$ so $\rho ={10}^{4\left({n}_{{\rm{\inf }}}-7\right)}.$ During the inflation, this energy density remains almost a constant; thus, ${n}_{{\rm{\inf }}}$ is invariant. Consider the future event horizon ${R}_{D};$ its formula is given by ${R}_{D}^{2}\sim 1/\left({{\rm{\Omega }}}_{{\rm{D}}}{H}^{2}\right)$ (this formula is derived in [1]). The Hubble ‘constant’ also remains invariant during the inflation, but ${{\rm{\Omega }}}_{{\rm{D}}}\sim {a}^{-2};$ thus, the event horizon grows fast.
In story 2, we will postulate an intermediate energy scale ${{\rm{\Lambda }}}_{m}$ under which the effective quantum field theory exists, above which spacetime is no longer classical; thus, together with the quantum spacetime there may be new physics. But whether this new physics can be ‘observed’ using ‘new apparatus’, is a good question. Now, if the inflation theory is correct, the UV cutoff of the inflation ${{\rm{\Lambda }}}_{{\rm{\inf }}}$ must not be greater than the intermediate energy scale ${{\rm{\Lambda }}}_{m},$ so that the effective field theory can be used. In story 3, we will argue that our new energy scale must be ${10}^{12}\,$GeV. The above is our prediction for the inflation.
In our universe, the e-fold number of the inflation is denoted by N, and according to observations, we have ${N}=60,$ so together with equation (1.4 ), we can infer
$\begin{eqnarray}{n}_{{D}_{i}}-{n}_{{D}_{f}}=30.\end{eqnarray}$
In the holographic dark energy model [1], there is a parameter c whose value is about 1. The almost exact formula for the inflation event horizon is ${R}_{{\rm{\inf }}}^{2}=\tfrac{3{c}^{2}{\bar{M}}_{p}^{2}}{\,{\rho }_{{\rm{\inf }}}\,{H}^{2}\,}={c}^{2}/\left({{\rm{\Omega }}}_{{\rm{\inf }}}\,{H}^{2}\right).$ In this paper we will always assume $c=1.$ Again, during inflation the Hubble constant remains almost invariant; thus, we have $\begin{eqnarray}\displaystyle \frac{{R}_{{D}_{f}}^{2}}{{R}_{{D}_{i}}^{2}}=\displaystyle \frac{{{\rm{\Omega }}}_{{D}_{i}}}{{{\rm{\Omega }}}_{{D}_{f}}}={{\rm{e}}}^{2N}={10}^{52},\end{eqnarray}$
where ${R}_{{D}_{f}}^{2}$ is the IR cutoff of the dark energy, which is different for the inflation horizon playing the role of the UV cutoff.To summarize, we gained the following from the story:
| 1. Equation ( |
Story 3 will tell us that the UV cutoff of the inflation is under a new energy scale of ${10}^{12}$ GeV
Story 2
For many and for me in particular, the most important evidence for the anthropic principle and the multi-verse scenario is that life exists in our universe. Life is a very complicated thing and the probability of the occurrence in a universe is very small. If there is a large number of universes, then life can appear in one of them, and our universe is such a universe.
It is interesting to note that the multi-verse theory in string theory or M-theory [7] is not explained well, for M-theory cannot explain the HDE yet.
In story 3, our gain will be that the new energy scale, namely, the intermediate energy scale in our universe must be ${10}^{12}$ GeV, this is to say ${n}_{m}=0.$
An important implication of holography is that if an observable universe has a finite size, then it is also a finite quantum system, and the number of all quantum states is given by $M=\exp \left(S\right),$ where S is the entropy of this finite universe.
But entropy S is given by the Bekenstein–Hawking formula, S $=\displaystyle \frac{A}{4},\,A=\pi {R}^{2},$ where R is the radius of the event horizon. After doing some simple calculations, we find that the entropy of our universe at the present time is $A\sim {10}^{120}.$ Thus, the number of quantum states of our universe at present is
$\begin{eqnarray}{M}\sim \exp \left({10}^{120}\right).\end{eqnarray}$
Of course, this is a huge number. This number, according to thermodynamics, must be greater than the entropy of our universe when the inflation stops. In story 1, one of the results is ${n}_{{D}_{i}}-{n}_{{D}_{f}}=30,$ and in turn, this formula implies that the IR cutoff when inflation starts is much smaller than the IR cutoff when inflation stops. So, the number of quantum states grows rapidly during inflation.Now, we provide some detailed calculations. The IR cutoff of inflation (not the dark energy, these two are very different) is almost invariant, namely, ${R}_{{\rm{\inf }}}^{2}\sim \tfrac{1}{{{\rm{\Omega }}}_{{\rm{\inf }}} \ {H}^{2}};$ then, the Bekenstein–Hawking formula tells us that2.2 ), we obtain2.3 )). But ${n}_{{\rm{\inf }}}\lt {n}_{m},$ the maximum of ${n}_{{\rm{\inf }}}$ is just ${n}_{m}.$ Substituting this result into equation (2.3 ), we find that the exponent of 10 is $4\left(7-{n}_{m}\right)-1,$ The minimum of this formula occurs when ${n}_{m}$ takes its largest possible number. In story 3, we will see that the largest possible value of ${n}_{m}$ is 0. Then, $4\left(7-{n}_{m}\right)-1=27,$ and we find that during inflation, the minimum of the holographic entropy is2.1 )). But equation (2.4 ) is still a very large number, and the corresponding number of all quantum states is
$\begin{eqnarray}S=\displaystyle \frac{\pi }{{{\rm{\Omega }}}_{{\rm{\inf }}}{H}^{2}}.\end{eqnarray}$
For ${{\rm{\Omega }}}_{{\rm{\inf }}}{H}^{2}=\tfrac{{\rho }_{{\rm{\inf }}}}{3{{\bar{M}}_{p}}^{2}}\sim 3\times 10\times {10}^{4\left({n}_{{\rm{\inf }}}-7\right)},$ where ${\bar{M}}_{p}$ is the reduced Planck energy scale, one order smaller than the Planck scale, substituting this result into equation ( $\begin{eqnarray}S\sim {10}^{4\left(7-{n}_{{\rm{\inf }}}\right)-1}.\end{eqnarray}$
During inflation, the inflation energy density is invariant, so we only demand that the IR size of inflation is greater than the corresponding space size of the new energy scale, namely, ${n}_{{\rm{\inf }}}\lt {n}_{m}.$ Apparently, when ${n}_{{\rm{\inf }}}$ grows, the holographic entropy of the universe decreases (equation ( $\begin{eqnarray}S\sim {10}^{27}.\end{eqnarray}$
This value is much smaller than entropy at present (equation ( $\begin{eqnarray}M=\exp \left(\,{10}^{27}\right).\end{eqnarray}$
Apparently, the conjecture, according to Penrose [8], is that our universe originated from a small quantum system is incorrect.Conclusions
Under the assumption that the HDE model is valid, then the effective field theory is also valid. If in our universe such a theory is the standard model plus the minimal axion model, then the driving force of the inflation, namely, the inflation is an axion. Before the PQ symmetry breaks, the inflation ‘vacuum’ is indeed cool, while all the entropy is contained in the holographic entropy of the dark energy. After the PQ symmetry breaks, the axion plays the role of dark matter; this is a story we will tell in the last story, story 3. In the reheating process, the entropy of matter will increase greatly. As for the huge quantum number of the present universe, it agrees with the anthropic principle [9].
Story 3
Why for some time I have believed that the minimal axion model is the ‘one’? From [10, 11], we know that the new energy scale agrees perfectly with the most natural axion model. Besides, the idea that axions form a superfluid is also very natural. In a superfluid state, axions are seen in the same quantum states. In such a situation, axions are described by a classical field $\phi ,$ whose stress tensor is3.1 ) and (3.2 ), we find that $\displaystyle \frac{1}{2}{({\partial }_{t}\phi )}^{2}=V,$ and that this scalar field must be locally homogeneous. The condition $\displaystyle \frac{1}{2}{({\partial }_{t}\phi )}^{2}=V$ leads to $\ddot{\phi }=\,{V}^{{\prime} }$ and this agrees with the e.o.m $\ddot{\phi }=\,-{V}^{{\prime} }$ only when ${V}^{{\prime} }=0,$ and this condition can be met approximately if we consider the time average of the oscillating axion field around the potential minimum (within this superfluid of dark matter, CP symmetry is violated).
$\begin{eqnarray}{T}_{\mu \nu }={\partial }_{\mu }\phi {\partial }_{\nu }\phi -\left(\displaystyle \frac{1}{2}{\partial }^{\mu }\phi {\partial }_{\mu }\phi +V\right){g}_{\mu \nu }.\end{eqnarray}$
As a fluid, its stress tensor assumes the form $\begin{eqnarray}{T}_{\mu \nu }=p{g}_{\mu \nu }+\left(\rho +p\right){u}_{\mu }{u}_{\nu }.\end{eqnarray}$
As a candidate of dark matter, $p=0,$ from equations (Next, we argue that this axion scale ${10}^{12}$ GeV ought to be the intermediate energy scale in our universe, that is to say, in our universe, any observable energy carried by a fundamental particle cannot exceed this scale. The arguments are the following:
| 1. If in our universe, the effective field theory is the standard model plus the minimal axion model, then any equipment used to detect fundamental particles is composed of some matter, whose components are within this effective field theory. Thus, any apparatus cannot detect any particle with an energy above this new scale. Therefore, we obtain a consistent result: the intermediate energy scale cannot be greater than ${10}^{12}$ GeV [10]. | |
| 2. An instability of the standard model occurs at the energy scale ${10}^{12}$ GeV [12], and if we add the minimal axion model, instability still occurs at this scale. |
Combining the above two points, we find that the effective field theory of our universe is valid under this new scale of ${10}^{12}$ GeV. Therefore, we claim that in our universe, the effective field theory is the standard model plus the minimal axion model. The axion field plays the role of the inflation during inflation, and after PQ symmetry breaking, it turns into dark matter.
Above the scale of ${10}^{12}$ GeV, we need a theory of quantum gravity, in other words, spacetime is no longer classical. Taking quantum spacetime into account, new physics may appear.
Here a few details are needed. We know that the so-called natural inflation driven by an axion has a flat potential when the decay constant ${f}_{a}$ is much larger than the Planck scale, if the axion potential is
$\begin{eqnarray}V\left(\phi \right)={{\rm{\Lambda }}}_{a}^{4}\left[1-\,\cos \left(\displaystyle \frac{\phi }{{f}_{a}}\right)\right..\end{eqnarray}$
However, as pointed out in [13], in certain situations, the axion can undergo monodromy so that the scalar field excurses a large value ${\rm{\Delta }}\phi \gg \,{f}_{a},$ and the axion potential is modified to $\begin{eqnarray}V\left(\phi \right)={\mu }^{4-p}{\phi }^{p}+{{\rm{\Lambda }}}_{a}^{4}\left[1-\,\cos \left(\displaystyle \frac{\phi }{{f}_{a}}\right)\right.,\end{eqnarray}$
where $p=\tfrac{2}{3},\tfrac{4}{3},$ as inflation proceeds along the $\phi $ direction, one has a slow roll on the ${\phi }^{p}$ piece, and is consistent with the observational constraints. Note that although we allow ${\rm{\Delta }}\phi \gg \,{f}_{a},$ the physics do not involve energy above ${f}_{a}.$Unfortunately, the potential (equation (3.4 )) is inconsistent with some recent observations of a positive $p$, and in particular, for $p=\tfrac{2}{3},$ although only at the 2σ level [14]. Fortunately, as discovered in [15], the coupling of the axion to other moduli generically makes the scale and the decay constant ${f}_{a}$ become functions of the axion. For example,3.5 ) is called the frequency drift.
$\begin{eqnarray}{f}_{a}={f}_{0}{\left(\displaystyle \frac{\phi }{{\phi }_{0}}\right)}^{-{p}_{f}}.\end{eqnarray}$
Equation (As studied in [16], when $p=\tfrac{2}{3},$ ${\phi }_{0}=8.78,\,{p}_{0}=-0.7$ (where ${\phi }_{0}$ is written in the unit of the Planck mass, and ${\,{\rm{\Lambda }}}_{a}$, our new energy scale, is unimportant, since inflation is dominated by the first term in equation (3.4 ), which is a scale beyond the Planck scale), this monodromy axion model is perfectly consistent with recent observations on the primordial gravitational waves, in particular with [14] and Planck data. This consistency between observations and the natural frequency drift, in my view, is a strong indication that string theory is truly behind the cosmic inflation.
As for the reheating process, it depends on the details of the axion model, but there is a universal term in the Lagrangian
$\begin{eqnarray}{g}_{\phi \gamma }\phi {F}_{\mu \nu }\tilde{{F}_{\mu \nu }},\end{eqnarray}$
where ${g}_{\phi \gamma }$ is proportional to the fine structure constant. Thus, during the reheating process, the inflation decays partly into axion dark matter as a complexified scalar $\phi $ settles into the potential minimum valley and partly into other particles through firstly decays into photons, at least.We need to stress that during the reheating process, although the superfluid of the axion as a decay product is homogeneous, there is no problem for later clustering, as the dark matter and other matter will become inhomogeneous later, as the small inhomogeneous generated by quantum fluctuations of inflation will drive them to clusters later. This is the cause of the general scenario of cosmic inflation.
Let $m$ be the axion mass, which is our model is typically much smaller than $\mu ev,$ the decay of axion to photons is given by3.7 ). For a more general discussion on the monodromy axion inflation and reheating, see [17], where it is found that an energy scale is ${10}^{12}$ GeV, which again serves as evidence of our scenario.
$\begin{eqnarray}{\rm{\Gamma }}\,\left(a\to \gamma \gamma \right)=\displaystyle \frac{4\pi {\alpha }^{2}{m}^{2}}{{{\rm{\Lambda }}}_{PQ}^{2}}{C}^{2},\end{eqnarray}$
where $C$ is proportional to ${g}_{\phi \gamma }$ and an energy scale, ${{\rm{\Lambda }}}_{PQ}$, is the PQ scale. With an appropriate choice of this coupling constant, we can get the right ratio of matter (photons) and the dark matter. Of course, the axion also decays into gluons, and the decay formula is similar to equation (Conclusions
In the three adventures we have described so far, we start with three assumptions:
| 1. Holography must be correct; therefore, the holographic dark energy model is correct. | |
| 2. There exists a new energy scale, 7 orders of magnitude smaller than the Planck scale, and above this intermediate scale, quantum spacetime appears. | |
| 3. The standard model plus the minimal axion model is the effective field theory in our universe. |
Through the three adventures, we finally have a closed logic chain. Three assumptions can be taken as a consistent axiom system. Thus, from this axiom system, we make the following predictions:
| 1. There is an intermediate energy scale, and the best choice is ${10}^{12}\,{\rm{GeV}}.$ Under this scale, effective field theory is valid. | |
| 2. Under this new scale, there is no new physics beyond the standard model plus the minimal axion model. The axion plays both the roles of the inflation and the dark matter. In between the new energy scale and the Planck scale, quantum spacetime already appears, and some new physics may appear. | |
| 3. Dark energy is the HDE model proposed in [1]. |
Perhaps, the third prediction, a prediction about the nature of the dark energy will be the hardest to confirm by future observations. But, confirmation is still possible in the not-too-far future, since the HDE model contains only one parameter c [1]. This parameter is close to 1. When it is 1, then dark energy will tend to behave like a constant. When c>1, the dark energy density will decrease. When c<1, the dark energy density will increase, and will become infinite in a finite future time [18, 19], namely, a big rip will happen.
The second prediction involves both an inflation model and axion detections. There are then two sub-predictions, and these predictions are relatively easier to confirm by astronomy observations and experiments in labs.
To summarize, in this paper we present a fairy tale of winter. The gains we obtain through the three adventures are hard to believe, but again I would like to believe.
Certainly, there is a lot of work to do to fit the data we already have: data about the cosmic microwave observations, data about the constraints on the minimal axion model, etc. I hope we will experience a period of excitement.