A fairy tale of winter -a theory about dark energy,dark matter,and inflation

Miao Li

School of Physics and Astronomy,Sun Yat-Sen University,Zhuhai 519082,China

Abstract This fairy tale begins with the journey of the famous Zhuang Zhou dreaming of a butterfly: I often do not know whether I am living in a dream,dreaming of a butterfly,or is a butterfly dreaming of me? Furthermore,there is the saying that those who believe it will see it,while for those who see it,it is then too late to believe it.The first thing I believe in is the holographic principle,and the holographic dark energy brought about by the holographic principle.Not until last year,I began to believe that there must be a new energy scale in between the Planck scale and the standard model scale.I also believe that this new scale is related to the minimal axion model.The results of the journey described in this paper are the following: 1.There is a new energy scale,which we will call the intermediate energy scale,and the best choice is 1012 GeV.2.The minimal axion exists,and under the new scale,there is no new physics.The axion plays both the roles of inflation and dark matter.New physics between the new scale and the Planck scale may exist because in this energy range,quantum gravity becomes important.3.The dark energy is holographic dark energy.Then all of the observable effects of all three predictions about dark energy,dark matter,and cosmic inflation are to be observed.

Keywords: holographic dark energy,new energy scale,standard model

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

when the inflation starts,at time 1,and when the inflation stops,at time 2.

According to equation (1.1),

Therefore,

Let ΛDbe the infrared energy cutoff corresponding to the dark energy,then ΛD=1012+nGeV=10n-7Mp=10n-7,where we set the Planck scale to be 1.For simplicity,we ignore the order one factors.Then we infer the following:

In story 2,we will postulate an intermediate energy scale Λmunder 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 Λinfmust not be greater than the intermediate energy scale Λm,so that the effective field theory can be used.In story 3,we will argue that our new energy scale must be 1012GeV.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 haveN=60,so together with equation (1.4),we can infer

whereis 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 (1.6) implies that in our universe when the inflation stops,it IR cutoff is much greater than the IR cutoff when the inflation starts.That is to say,if the HDE model is valid,then the dark energy future event horizon is much greater than its size when inflation starts;this agrees perfectly with the second law of thermodynamics.

Story 3 will tell us that the UV cutoff of the inflation is under a new energy scale of 1012GeV

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 1012GeV,this is to saynm=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 byM=expS(),where S is the entropy of this finite universe.

This value is much smaller than entropy at present(equation (2.1)).But equation (2.4) is still a very large number,and the corresponding number of all quantum states is

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 fieldφ,whose stress tensor is

As a fluid,its stress tensor assumes the form

If the minimal axion model accounts for the dark matter,then its energy scale cannot exceed 1012GeV,as argued some time ago in[10].For a further discussion of axions,see [11].

Next,we argue that this axion scale1012GeV 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 1012GeV [10].

2.An instability of the standard model occurs at the energy scale 1012GeV [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 1012GeV.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 1012GeV,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 socalled natural inflation driven by an axion has a flat potential when the decay constantfais much larger than the Planck scale,if the axion potential is

However,as pointed out in [13],in certain situations,the axion can undergo monodromy so that the scalar field excurses a large valueΔφ≫fa,and the axion potential is modified to

Equation (3.5) is called the frequency drift.

As studied in[16],whenp=,φ0=8.78,p0=-0.7(whereφ0is written in the unit of the Planck mass,and Λa,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

wheregφγis proportional to the fine structure constant.Thus,during the reheating process,the inflation decays partly into axion dark matter as a complexified scalarφsettles 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.

Letmbe the axion mass,which is our model is typically much smaller thanμev,the decay of axion to photons is given by

whereCis proportional togφγand an energy scale,ΛPQ,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 (3.7).For a more general discussion on the monodromy axion inflation and reheating,see [17],where it is found that an energy scale is 1012GeV,which again serves as evidence of our scenario.

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 1012GeV.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-toofar 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.

Acknowledgments

For a long time,I have been influenced by Prof.Chun Liu,and began to believe that the standard model is valid in our universe.Not until recently,I started to believe that the standard model plus the minimal axion model is the effective field theory of our universe,and there ought to be a new energy scale.I am grateful to Prof.Han-Qing Zheng as he told me that 1012GeVis the scale at which the standard model vacuum becomes instable.I am grateful to Prof.Wu-Ming Liu for his Wechat group where I have benefitted from many discussions.Lastly,I thank the referees for the critical reading of this paper and useful suggestions.This work is supported by a grant from the National Natural Science Foundation of China: 12175099.