An Exact Scalar Field Inflationary Cosmological Model Which Solves Cosmological Constant Problem , Dark Matter Problem and Other Problems of Inflationary Cosmology

An exact scalar field cosmological model is constructed from the exact solution of the field equations. The solutions are exact and no approximation like slow roll is used. The model gives inflation, solves horizon and flatness problems. The model also gives a satisfactory estimate of present vacuum energy density as well as vacuum energy density at Planck epoch and solves cosmological constant problem of 120 orders of magnitude discrepancy of vacuum energy density. Further, this model predicts existence of dark matter/energy and gives an extremely accurate estimate of present energy density of dark matter and energy. Along with explanations of graceful exit, radiation era, matter domination, this model also indicates the reason for present accelerated state of the universe. In this work a method is shown following which one can construct an infinite number of exact scalar field inflationary cosmological models.


Introduction
Inflation was proposed by Alan Guth [1] although the idea of an exponential type expansion was due to Starobinsky and others [2][3][4].The modern form of inflationary cosmology is due to A. Linde, A. Albrecht and P. Steinhardt [5,6].In Guth's original model the inflaton field was assumed to be trapped in a false vacuum and assumed a local value which is minimum.The inflaton field comes out from the local minimum value by quantum tunnelling and as universe inflates, tunnelling takes place.However, these ideas when pursued gave empty universe and therefore rejected.Guth further tried to improve the idea but they led to other difficulties.

 
Linde and Steinhardt proposed new inflationary model where the inflaton field varies slowly and undergoes a phase transition of second order.New inflationary models do not require the idea of tunnelling.Most of the modern models depend on the idea of chaotic inflation due to Linde.In these models the initial value of the inflaton field is set chaotically when the universe exits from Planck era.The field then rolls downhill and if the potential is enough flat then inflation can take place.
There are another class of models known as hybrid inflationary models in which two fields are considered.These models introduce extra difficulties but they can speculate some features of single field models.
Inflationary cosmology is important because it offers solution to some great puzzles of cosmology.The puzzles are Flatness problem, Horizon problem and Monopole problem.
Flatness problem is basically why the density parameter is extremely close to unity i.e. why Ω ≈ 1? Horizon problem is why the universe is extremely smooth and isotropic on large scales?Monopole and the unwanted relics are the problems associated with standard hot Big Bang Theory.They are trivially solved when Flatness and Horizon problems are solved.
The above problems namely Flatness problem and Horizon problem are problems of Standard Big Bang theory are solved by assuming an accelerated expansion in early universe for a very short duration.This accelerated expansion is named as inflation.The starting time of inflation is model dependent.However, it occurred when the universe was extremely young.Inflation ended around the time when universe was 10 -33 sec.old.From this time (10 -33 sec.)radiation domination started.The phenomenon of ending inflation and then entering into radiation dominated era is known as graceful exit.And its mechanism requires explanations.An entirely different mechanism of graceful exit will be given in this work.
Lot of scalar field inflationary cosmological models have been proposed so far to explain the above scenarios.Expansion of universe is assumed to be driven by a scalar field and an associated potential .Many forms of potentials [7][8][9][10][11] have been used to solve the associated field equations.
In some models a kind of approximation is used to solve the difficult equations.This approximation is known as slow roll approximation which assumes that the field rolls very slowly.Mathematically this is equivalent to assuming where the overhead dot represents derivative with respect to time.A few models find exact solutions to the field equations.All the above models explain the mechanism of inflation and solve Horizon and Flatness problems.Further it is found that solution of these problems are equivalent to produce an e-folding [defined as ln

The Scalar Field Equation and Its Exact
W xists a scalar are values of scale factor when inflation starts and ends respectively.
However these above models fail to explain cosmological constant problem [13] and dark matter problem.The cosmological constant problem is why the measured vacuum energy density is small by a factor of about from its theoretical value.This is in language of Weinberg; "Worst failure of an order of magnitude estimate in the history of physics".
The dark matter problem is another unsolved puzzle in modern cosmology.Our present knowledge asserts that the energy density of matter/energy content [14] of our universe is: dark energy ~ 74%, dark matter ~ 22% and ordinary matter ~ 4%.No cosmological model predicts or accounts for this observation.There is also the problem of present acceleration [15][16][17] of the universe found from the observation of distant Supernovae Ia.The present work addresses all the above problems listed from the beginning and provides solutions in a single framework.Further, the solution of cosmological evolution equations are exact and no sort of approximations like slow roll approximation etc. is used to derive the solutions.
It may be mentioned here that slow roll is not the necessary and sufficient condition of inflation.However, if slow roll is valid, inflation takes place.It will be shown in this work that without slow roll one can have plenty of exact inflationary models.
Most vital idea in this work is the attribution of negative energy density to dark matter/energy constituents tion of state of dark matter/energy.It is shown that this idea fitted in an exact mathematical framework essentially solves all problems of standard cosmology.

Solutions
e suppose that after tunnelling there e field  and an associated potential   V  , which is respon ble for the evolution of the univer It is further assumed that initially there existed some other type of fields i si se.

 
i i X  .But these fields were h ed up initially wh s ang ich mean  are negligible and they did not contribute to field equations initially.The number and nature of the i  fields are not important for the purpose of cosmological predictions.The interactions of the scalar field  with other fields are assumed to be ignorable and c sequently the i on  fields are assumed to interact among themselves onl Now if the inflaton fie y. ld  has no spatial variation an en d depends only on time th we can write the equations of motion [18] of the scalar field and the Friedmann equation ignoring the curvature term as: and where a is the scale factor,  is the inflaton field and

 
V  is the potential.Over ad dot represents derivaith respect to time and overhead prime represents derivative w.r. to he tive w  .Equation (1) follows from the Lagrangian [18]      Solution of Equations ( 1) and (2) are in some ways si one can find ex endix A) one can find as milar to the solution of Diophantine equations in Classical Algebra [19], where the number of unknowns are more than the number of equations given.
Here a method will be shown by which act solution of Equations ( 1) and (2).In principle we will choose an arbitrary function from which we can construct some form of potentials for which Equations ( 1) and ( 2) are exactly solvable.
Following this method (App m owing the method derived and illustrated in A any as exact solutions as one wishes.(In principle this method allows one to find an infinite number of exact solutions.)Now foll ppendix A, we write the solutions of (1) and (2).They are: (The overhead dot represents time derivative.)

  
The functions   f t is arbitrary so that one can have an r infinite numbe hoices of   of c f t and can have an infinite number of exact solutions.

The Exact Scalar Field Model and
Fr illustrated in Appendix A, we can now n:  

Solution of Flatness and Horizon Problems
om the method find an exact inflationary model.
We choose the arbitrary functio The results are (Appendix A) And the same potential [A.34] in Φ dependent form is: In this model we choose the starting time of inflation Now we take Then using (11a) and (11b) we obtain from (11) Therefore the e-folding one pe

Graceful Exit and Starting of Radiation
were negligible so that they did not ntribut field equations.When inflation starts the inflaton field decays.During the period of inflation particle production due to decaying inflaton field is assumed to be negligible and not taken into account.But all of the hanged up fields interact among themselves and produce new particles with significant negative energy density around the time .The exotic particles will be identified as d nergy later on.The natures of the particles depend on the theory concerned and their natures are not very important for our purpose.We take it for granted that many forms of exotic particles were formed around the time lt inflation must stop.The appearance of an overall condition 3 0 P    guarantees creation of a retarded phase mption of negative energy density particles is [18].The assu pe tive energy density due to cr rfectly consistent with the Null energy condition and Strong energy condition [13]. The where e neglect further variations of ã t .This is, however unnecessary for our purpose.

Af of universe continues
Matter/Energy Problem ter graceful exit the expansion and the inflaton field  goes on decaying.We assume that particles are produ ed in this phase with properties 0 P  , 0 c   as well as 0 P  0   .For the second typ f pa s if we assum ua of state e o rticle e an eq tion for these particles.All energy con ermit this ditions p [13].We take it for granted that these type of particles are produced more than the first type in matter dominated phase.Now 0    , since 0   for both type of particles.The r effect is the appearance of a positive energy density denoted by ove all I   .Thus total energy density of all created particles after graceful exit upto present moment is represented by With this idea we can now write mann equatio the Fried n at present epoch: Equation ( 15) follows from ( 14) by term (14)  Here   is the energy density of the inflaton field.

 
Now we define cosmological const density of the in field (i.e.
ant as the energy flaton Using (A.29a) and (A.29) we write is rgy density of dark matter/ener calculate its Equ the ene gy and present value.The negative sign before *   in (16)   indicates that energy density of dark matter/energy is negative.

65
Using (19) and ( 24) we find the present value of energy density of dark matter/energy as  matter/energy   [12].
Thus the present energy den get of the finds its correct accounting, 95.78% corresponds to dark m Now during the course of evolution, after graceful exit rther formatio nergy den-sity bud universe atter and energy and 4.22% corresponds to ordinary matter and energy.However there is a basic difference in the nature of the above energy densities.The energy density of inflaton i.e. vacuum energy density is positive, while the energy density of dark matter/energy is negative.The present energy density of ordinary matter-energy equals present vacuum energy density less the magnitude of present energy density of dark matter/energy.And as energy density of exotic particles were taken negative, it turns out that constituents of dark matter/ energy are exotic particles as energy density of dark matter/energy is also negative.

Matter Domination and Present Accelerated State of the Universe
It was explained in previous sections that the mechanism of graceful exit is due to formation of some kinds of particles due to interaction of the hanged up fields between themselves.
the energy density slowly increases due to fu n of new particles.Unlike exotic particles e sity, these particles have positive energy densities.So that they add up with inflaton energy density   .Cooling also increases of the energy density of the universe.And due to this overall increase of energy density, the universe gradually enters into matter dominated phase, when formation of matter takes place.
Present accelerated phase is due to further continuation of above features, i.e. formation of more and more positive energy density particles together with cooling etc.It was assumed in Section 5 that particles produced after graceful exit has the property 0

Summary and Concluding Remarks
A variety of cosmological models were proposed in last three decades to solve the major problems of cosmology.Among these are the Coleman-Weinberg SU (5) model, models by Pi [20] and Shafi and Vilenkin [21] and many other models.All the above models were either a failure or partially succes explain .sful to few features only.And ysteriet urther, th all models so far proposed failed to explain the m ous cosmological constant problem.No model has y predicted the existence of dark matter and energy.
The present work solves the mysterious cosmological constant problem i.e. the discrepancy of 120 or more precisely 122 orders of the measured value of cosmological constant and predicts the existence of dark matter and energy.The work removes the ambiguity of definition of cosmological constant by clearly defining it as scalar field energy density or vacuum energy density and not the energy density of dark matter/energy.F is model gives extremely accurate estimate of present values of vacuum energy density and energy density of dark matter/energy.It also solves flatness and horizon problem, gives a satisfactory estimate of e-folding which is necessary to solve horizon and flatness problems and of course trivially monopole problem.Lastly this work also supplies the explanation for the present state of acceleration of the universe.
This work although explains the major problems of present day cosmology, it is not clear whether this exact model will be able to explain far late behavior of our universe.And certainly it is not capable to predict any new cosmological phenomena which may occur in future.
The above work is a revised version of a work by this author [22].
So that from (A.6) and (A.11) one obtains (Negative sign is considered for convenience.)Next we find from (A.13) and (A.16) The above calculations assure that the exact solution of (A.1) and (A.2) can be found from the following prescription: One can check that (A.21) is the exact solution set of (A.1) and (A.2) in the following way: From the last of (A.21) one gets From the 2nd of (A.21) one obtains It is now easy to verify from (A.23) that  Now we will construct an exact inflationary model from the exact solutions obtained before.
Let us choose the arbitrary function Here A and B are real arbitrary constants i.e.

 
   Thus it turns out that the potential given by (A.34) produces the scale factor given by (A.27).The potential given by (A.34) and (A.29) are the same potential in different forms.

e e
an alternative possibility permitted by the equa-

5 N
If we choose A = 7.5 Then from(12) the res .


from the interaction of the hanged up fields wh tence were postulated earlier.In analogy with dark energy equations of state we take the equation of state of these particles as P    with  negative.However, we take the first p discussed before i.e. we take 0 P  and 0 o lity ssibi   for these exotic particles.And ap- ce of a negative energy density field helps to stop inflation large number of exotic particles.
indicates that the energy densities of the e tic particles are negative.At the time of graceful exit the universe enters into a de xo celerated phase.It is well-known[18] that the conditions of accelerated phase is 3 sume that the creation of new energy density due to newly born particles create an overall situation where an overall condition like 3 0 P .We can    holds from the time of graceful exit.The foregoing discu trate the mechanism of gr ssions illus n exactly aceful exit.An accelerated expansion reduces to a time half power law at the time of graceful exit i.e.


defined by ation ( In view of Equation (27) we can saf 95.78% energy density of the inflaton field ely conclude that is diminished by aterg 22% represent ordinary matter energy, since for ordinary 0 the presence of negative energy density of dark m ter/en y and the rest 4.

.
represent u stab e particles which have vacuum like properties.Now in matter dominated phase as more and more particles are produced with p And accelera f the universe starts right from the moment when 3P tion o     becomes negative.Such a situation still ues for which we observe our universe accelerating presently.It is once again mentioned that particles produced in various phases after graceful exit has

(
Taking positive sign of square root only).It has to be remembered that  f t is arbitrary.Then from (A.14) and (A.15).
.33) gives the  dependence of the po- tential which in more compact form can be recasted as , o A  ), and B > 0, Friedmann equation assumes a new form from the time of graceful exit.Considering the appearance of negative energy density particles we find that Friedmann equation (i.e.Equation (2a)) assumes its new form at the time of graceful exit: Equation (A.29) gives the time dependent form of the potential which gives the scale factor (A.27).Next we will find the scalar field  dependence of the potential in the following way:From the last of (A.21), we have