A unified cosmological model for solution of problems of Cosmological constant, Cosmic coincidence, Energy conservation, Flatness and homogeneity of Horizon

Cosmological constant problem is the difference by a factor of ~10 123 between quantum mechanically calculated vacuum energy density and astronomically observed value. Cosmic coincidence problem questions why matter energy density is of the same order as the present vacuum energy density (former is ~32% and latter is ~68%). Recently, by quantizing zero-point field of space, we have developed a cosmological model that predicts correct value of vacuum and non-vacuum energy density. In this paper, we remove some earlier assumptions and develop a generalized version of our cosmological model to solve three more problems viz. energy conservation, flatness and horizon problem along with the above two. For creation of universe without violating law of energy conservation, net energy of the universe including (negative) gravitational potential energy must be zero. However, in conventional method, its quantitative proof needs the space to be exactly flat i.e. zero-energy universe is a consequence of flatness. But, in this paper, we will prove a zero-energy universe without using flatness of space and then show that flatness is actually a consequence of zero energy density. Finally, using our model we solve the horizon problem of universe. Although cosmic inflation can explain the flatness of space and uniformity of horizon by invoking inflaton field, it cannot predict the present value of vacuum energy density or matter density. But, our cosmological model solves in an unified manner all the above mentioned five problems viz. cosmological constant problem, cosmic coincidence problem, energy conservation, flatness and horizon problem.


Introduction
The human civilization has been curious to know the nature and evolution of the universe since ancient times. Long back, our observable universe with three spatial dimensions was thought to be infinite in extent and was considered to exist since infinite time. But, modern science has concluded that the universe has a finite extent and finite age. This is due to the following reason. If the universe was infinite and there is a finite density of stars, while looking at any direction, our view must hit a star. So, we will be surrounded by an environment whose temperature is same as that of stars. But, this is not the case. Similarly, if the universe existed since infinite time, all parts of it would have reached thermal equilibrium (or same temperature) by now by exchanging radiations. But, we do see different temperatures on different objects in the universe. So, we have now concluded that the universe has a finite extent and finite age. Consequent to discovery of General theory of Relativity by Einstein and observation of expansion of the universe by Hubble, Big-Bang model was proposed in which universe was assumed to be created by a big explosion of radiation and matter and its age was estimated to be ~ 14 billion years. However, Big-Bang model of universe suffered from five major problems viz. Cosmological constant problem, Cosmic coincidence problem, Energy conservation, Flatness and Horizon problem. At first, we will briefly explain all these problems and then proceed to develop a unified cosmological model to solve them.

Cosmological constant problem and cosmic coincidence problem:
Our universe is expanding. From the observed rate of expansion of the universe and since space is observed to be flat, solution of Einstein's field equations i.e. Friedmann's equation indicates that total energy density due to matter, radiation and vacuum must be where H is Hubble's constant. On measurement, we find that matter energy including dark matter can account only ~32% of this critical density.
Radiation energy is found to be less than 0.1%. So, it is estimated that rest ~ 68% of the total energy of the universe must be vacuum energy (dark energy). Numerical value of this vacuum energy density comes out to be 5.3×10 -10 J/m 3 . However, as shown by Weinberg (1989), quantum mechanically calculated vacuum energy density comes out to be 3 114 4 2 Another significant problem in cosmology is the question, why is the matter energy density of the same order as the vacuum energy density today? This is surprising as matter energy density decreases as 1/R 3 where as vacuum energy density is assumed to be constant. This problem is known as cosmic coincidence problem [3].

Problem of conservation of energy:
Presently total matter (including dark part) in the universe is estimated to be about ~10 54 kg. The question is, what was the source of such a huge amount of energy? A quantum vacuum fluctuation can at best create Planck energy of ~0.02mg in a Planck time of ~ 5.4×10 -44 sec. To answer this question, it was proposed that sum total of energy of the universe due to matter/vacuum and gravitation potential energy might be zero so that the law of conservation of energy of the universe is satisfied since its creation. This is possible as the sign of gravitational potential energy is negative whereas that of other forms is positive. To make net energy zero, magnitude of gravitational potential energy must be equal to the magnitude of non-gravitational part (matter plus vacuum). But, mathematical proof of zeroenergy universe needs the flatness of space as an assumption [4] i.e. zero-energy universe is a consequence of flatness. Since the space became flat only after inflation, the question remains about the source of energy before and during inflation. But, in this paper, we will prove zeroenergy universe without using flatness of space and so, energy can be conserved right from the beginning of the universe. In our model, flatness is actually a consequence of zero net energy density in the universe.
Of course one can argue, in an expanding space-time, Einstein's field equation doesn't imply conservation of energy due to matter and radiation [5]. This happens because Einstein's theory of gravity is a geometric theory and it considers only matter and radiation in its energy-momentum tensor. But, if we bring into the picture the concept of gravitational potential energy and are able to develop a cosmological model that satisfies the law of conservation of energy, the model will be scientifically more satisfying.

Flatness and Horizon problems and inflationary solution:
Infinitesimal length of a line element in our space-time is given by the generalized Friedmann-Robertson-Walker (FRW) metric, Where 'R' is scale factor (or radius of curvature), k is observed curvature constant which can be +1 for positive curvature, 0 for flat space or -1 for negative curvature. Note that k can take any value (+1, 0 or -1) depending upon the total energy density and rate of expansion of the universe even if the universe is physically closed (like an observer freely falling in a gravitational field cannot detect the curvature of space-time). Thus, in addition to energymomentum tensor, dynamics of the observer plays a very important role in the curvature measured by him. Friedmann's solution of Einstein's field equation is given by [6][7], Critical density is defined by the dynamical properties of the universe is, G Only if  is exactly equal to 1, then k = 0. If >1, k = +1. If <1, k = -1.
As in a matter dominated universe,  varies as R -3 , Eq. (4) indicates that deviation of  from 1 is expected to increase in direct proportion to R. This instability of  near the value of ~ 1 is known as Flatness problem. To be specific, presently  is found to be close to 1 with an accuracy of better than 0.5%. As R has increased by a factor of ~10 61 (from Planck size~10 -35 m to current size ~10 26 m),  must be close to 1 in the initial time with an accuracy of better than 10 -62 %. Thus, flatness problem seeks an answer for such a precise fine tuning of  at the initial stage of the universe.
Horizon problem poses a challenge to explain the extraordinary homogeneity observed in temperature of causally disconnected regions of the universe despite of the finite speed of light. With expansion of the universe, temperature of the universe decreased and at about ~380,000 years, temperature was sufficiently low for recombination of electrons and protons and thus neutral atoms were formed. As neutral atoms have very low cross section for scattering of photons as compared to charged particles, photons decoupled from the plasma and have been streaming through the universe since that time. Of course, due to expansion of the universe, their wavelengths have been stretched and these photons are known as Cosmic Microwave Background (CMB) radiation. When we measure the spectrum of this CMB radiation, it very well fits to a black body spectrum of temperature of ~2.726 K. Even if we compare the temperatures of two opposite regions of the sky by their CMB radiation spectrum, both the temperatures agree with each other within an accuracy of ~0.001 K. The fact that we are measuring 'now' indicates, the two opposite regions have just come in causal contact with us. So, they must not have been in causal contact with each other as radiation from one part has covered only half the distance from other part. In fact, following the method of [8], it can be shown that the observable universe consists of about 36000 causally disconnected patches which surprisingly have almost same temperatures. This is as follows.
If the present radius of the universe in form of 3-sphere is R 0 , the co-moving particle horizon at any time t is given by, Taking R=At  (for matter dominated universe =2/3 and for radiation dominated =1/2), If t is the time when photon decoupling occurred releasing the CMB radiation and t 0 is present time, the ratio of co-moving horizon volume today to the co-moving horizon volume at photon decoupling is, Where z=Red shift of CMB radiation.
Still, they have almost same temperature of ~2.726 ± 0.001 K. This is known as Horizon problem.
In order to explain the Flatness and Horizon problems, Alan Guth [9][10] and others have proposed the cosmic inflation theory which assumes the existence of an inflaton field having a large potential energy in the early universe. Its negative pressure caused exponential expansion of the universe from ~10 -30 m to ~10 -2 m during a period from about ~ 10 -35 s to ~ 10 -32 s after creation of the universe. Since the energy density of the inflaton field is assumed to be nearly constant during this expansion, Eq. (4) indicates that whatever initial value  starts with, it approaches 1 with increasing radius R (as deviation from 1 is proportional to R -2 ). Thus, instability of  is avoided during early stage of universe. After ~10 -32 s even if inflation stops,  is already so close to 1 that it will take very long to deviate from 1. This is how inflation theory explains the avoidance of instability of  in early universe and why the value of  is ~1. Similarly, as the universe expanded by a huge factor of ~10 28 in a very small time interval of ~10 -32 s, outer portion of every causal patch was pushed out of the horizon. After inflation stopped and as horizon still expands at the speed of light, we are seeing those parts which were originally in causal contact with us before inflation. Thus, horizon problem is solved in cosmic inflation theory.
But, the limitation of the inflation theory is that it cannot explain the present rate of expansion of the universe by predicting the value of vacuum energy density and it also can't predict matter energy density. Since the space became flat only after inflation, inflationary model can't prove the net energy of universe to be zero before inflation and thus the law of energy conservation is violated during pre-inflation period. In addition, the magnitude and variation of the potential energy with strength of the inflaton field are artificially decided to make it fit to explain the flatness and horizon problems. That's why currently there are so many inflationary models proposed in literature. Mechanism of graceful exit from inflationary stage also casts some doubt on the theory. Recently due to this, Paul J.
Steinhardt, one of the architects of the theory, has turned against it [11][12].

Our approach:
To solve the above mentioned problems in an unified manner, we will develop a generalized version of our recent cosmological model [13] where we derived the expression to calculate the density of vacuum energy that needs only Hubble's constant as input (whereas standard cosmology needs both Hubble's constant and matter density). We had also developed expression for energy density of matter component for which standard cosmology has no predictive power. However, earlier we had taken some assumptions which led to approximate results. In this paper, we will remove those assumptions and develop a more generalized version of cosmological model that explains how creation of universe doesn't violate the law of conservation of energy and also simultaneously solves the cosmological constant problem, cosmic coincidence problem, flatness problem and horizon problem of the universe.
The structure of the paper is as follows. In the next section, we describe our improved version of cosmological model that leads to expression for vacuum and matter energy density and solves cosmic coincidence problem. Using these expressions, in the next three sections, we demonstrate working of the law of conservation of energy and then solve the cosmological constant, flatness and horizon problems in our model. Lastly, in the conclusion, we present a table comparing the advantages and predictive power of our cosmological model with that of the standard model.

Development of generalized cosmological model and solution of cosmic coincidence problem
In standard cosmology, universe is assumed to be isotropic and homogeneous. To impose the condition of isotropy, Einstein visualized the universe as a closed hyper-surface (surface of a 3-sphere) existing in a 4-dimensional space. This exciting idea of Einstein as expressed in his original paper [14] is quoted below.
"We arrive at such a space, for example, in the following way. We start from a Euclidean space of four dimensions,  1 ,  2 ,  3 ,  4 , with a linear element d; let therefore, In this space we consider the hyper-surface, To maintain isotropy of space, we will follow the Einstein's model of universe that lies on the surface of a 3-sphere (hyper-sphere) existing in a four-dimensional space. As described in section-1, curvature constant k in FRW metric of this space represents the observed characteristics of a line element depending upon energy density and Hubble's constant. So, even if globally the universe is closed and finite, k may be +1, 0 or -1 depending upon the rate of expansion of universe (just like space-time looks flat to an observer freely falling under a gravitational field). As per quantum field theory, this space in the form of hypersurface of 3-sphere must have the zero-point energy and sum of energies for all possible modes of oscillations is the vacuum energy. We will assume a common zero-point scalar field that is responsible for energy of all the fundamental particles. Number of particles having same energy increases when a particular zero-point mode is excited. Properties of particles other than energy appear when excited field couples with other fields such as Higg's field, electromagnetic field etc. So, particles of same energy even if have different rest mass, charge or spin correspond to a single and common mode of zero-point wave.
Since the space of our universe is closed and spherically symmetric, circumference must be integer multiple of wavelength of zero-point wave. Mathematically, Where R is the radius of universe in the form of 3-sphere,  is wavelength of zero-point field, n is an integer varying from 1 to n max and n max is decided by lowest possible wavelength  min .
Minimum wave length must be of the order of Planck length , as distances smaller than Planck length cannot be defined [15][16]. We will take the cutoff wavelength as where  is a constant nearly equal to 1. Using this min  in Eq. (7) we get, Initial radius R i of universe can be found from Eq. (8) by putting max n equal to 1 and it is given by, Let us now consider the example of a particle trapped in a one-dimensional box of size L. Although its magnitude of momentum is quantized by, As R >> R i , second term in (10) and (11) is negligible and so, vacuum energy and its density are given by, Numerical value of vacuum energy density vac u will be calculated in section-4 after we derive the analytical expression for R.
Now, Let us calculate the average energy density due to non-vacuum components (ordinary matter, dark matter and radiation) of the universe. If  is scalar field of the zeropoint oscillation, energy density and pressure are given by [17], Just like in a harmonic oscillator, energy will be shared equally between potential energy (depending on ) and kinetic energy (depending on derivatives of ). So, Using Eq. (16) and (17) in Eq. (14) and (15), we get, Expanding the bracket in Eq. (10) and then taking differential, In a matter dominated universe, pressure is mainly caused by vacuum energy (gravitational potential energy and its pressure are not considered in Einstein field equation as it is a geometric theory and pressure due to radiation is neglected as radiation energy is less than Thus, we find that non-vacuum energy of the universe also increases with radius of the universe. Increase of non-vacuum energy happens by way of excitation of zero-point oscillators (i.e. creation of matter and radiation). We will prove in next section that the energy required for increase of both vacuum and non-vacuum parts comes from gravitational potential energy which becomes more and more negative and keeps the net energy of universe zero.
Integrating Eq. (21) from R i to R, we get,  Thus, the cosmic coincidence problem which questions why the matter energy density is of the same order as vacuum energy density is explained.

Conservation of energy in an expanding universe
Using Eq. (13) and (25), total mass density due to vacuum and non-vacuum energy is given by, in Fiedmann's Equations (2) and (3), we get, Thus, we find that the radius of universe increases with time at a uniform rate of . So, at any instant of time , radius of universe is, After , which is Newton's law.
So, we can safely use Newton's law for calculating gravitational potential energy. However, since gravitational effect is transmitted at the speed of light, we will use the retarded form of Newton's expression for gravitational potential energy and also consider time dependence of energy density of the universe. As shown in Fig. 1, mass at a specific time  contained in thin strip on the surface of hyper-sphere at angular distance of  from a unit mass can contribute to the gravitational potential energy at any time t and distance to be used in Newton's expression must be c(t-).
Another condition is that the  must be equal to the angular distance covered by the gravitational radiation during the time interval from  to t. Mathematically,

Solution of Flatness problem and Cosmological constant problem
We first restate the Eq.
Thus, as deviation of  from 1 remains constant irrespective of the radius of the universe, the instability of  (which is main concern for flatness problem) is eliminated. However, as  is a constant as per Eq. (33), for three possible values of k = +1, 0 and -1, there are three possible values of . We will now predict the expected value of  which will be proved to be equal to 1.
In the previous section we have proved that law of energy conservation can be satisfied for specific pairs of values of  and k. When it happens, gravitational potential energy density u grav becomes equal and opposite of the average non-gravitational energy density u matt+vac (matter plus vacuum Second term in right hand side of above equation represents the contribution due to gravitational field in the total energy-momentum tensor. In our case, we have seen that contributions of gravitational field make both total energy density and pressure equal to zero. Thus, as the right hand side of Eq. (34) i.e. total energy-momentum pseudo-tensor becomes zero all over the space of the universe, curvature of space-time also becomes zero i.e. k = 0.
Putting k=0 in k   Finally, using the conclusion of previous section i.e.  = 2 (corresponding to k=0) for energy conservation, rate of expansion of universe is,

Solution of Horizon problem:
As the rate of expansion of universe is constant i. Then, the ratio of co-moving horizon volume at present time t 0 to the co-moving horizon volume at the time t CMB of photon decoupling is, , t 0 =14.4 by and t CMB =380000 y, Thus, we find that, in contrast to standard cosmology, the whole observable universe is nearly causally connected and it doesn't contain ~36000 causally disconnected patches.
We will also give here an alternative calculation which will show that the co-moving size of patch from which CMB photons come to us is much smaller than the co-moving particle distance at the time of release of CMB radiation. After the release of CMB radiation, the photons we are detecting now must have moved through an angular distance of, Putting t 0 =14.4 by and t CMB =380,000 y,  = 5.27 radian. Since, this is more than , as shown Theoretically maximum angular distance a photon can move till the time of photon decoupling is, i.e. maximum angular distance that a photon can move till CMB release is about ~33 times more than the angular distance between two opposite points of horizon, it is possible for our horizon to attain thermal equilibrium and achieve nearly same temperature.
Thus, our cosmic model solves the horizon problem.

Conclusion
In this paper, we have developed a unified cosmological model to solve five important astrophysical problems viz. cosmological constant problem, cosmic coincidence problem, energy conservation, flatness and horizon problem. Our model predicts the density of vacuum energy taking only Hubble's constant as input whereas standard cosmology needs both Hubble's constant and matter density to estimate vacuum energy density. We have also developed expression for energy density of non-vacuum component (matter pus radiation) which standard cosmology can't predict. In contrast to standard inflationary model, our theory doesn't need flatness of space as a pre-requirement to prove zero-energy universe or to save law of conservation of energy. In our model, flatness of space is a consequence of zeroenergy universe. We have found that total vacuum and non-vacuum energy increases with expansion of the universe and it comes from gravitational potential energy which becomes more and more negative. In contrast to standard cosmology, our model doesn't have to invoke any inflaton field having a specific potential energy distribution to prove flatness and homogeneity of horizon. It just uses quantized zero point scalar field in a closed universe.
Highlights of the advantages and predictive power of our cosmological model as compared to standard model are given in Table. 1.
At present, standard ɅCDM model is used to infer evolution of temperature and density of universe from CMB spectra and anisotropy. But, if we take same CMB spectra and anisotropy data and apply our new cosmological model, we expect to reach at somewhat different conclusions about temperature and density of early universe. This part along with nucleosynthesis and galaxy formation will be investigated in future.