Open Astron. 2019; 28: 220–227

Research Article

Biswaranjan Dikshit* Mechanical Explanation for Dark , Cosmic Coincidence, Flatness, Age, and Size of the

https://doi.org/10.1515/astro-2019-0021 Received Sep 27, 2019; accepted Dec 16, 2019 Abstract: One of the most important problems in is the problem in which conventional calculation of energy using quantum leads to a value which is ~10123 more than the estimated from astronomical observations of expanding universe. The cosmic coincidence problem questions why energy density is of the same order of as the vacuum energy density at . Finally, the mechanism responsible for spatial flatness is not clearly understood. In this paper, by taking the vacuum as a finite and closed quantum oscillator, we solve all of the above-mentioned problems. At first, by using thepurely quantum mechanical approach, we predict that the density is c4/(GR2) = 5.27×10−10 J/m3 (where R is radius of 3-sphere of the universe) and matter energy density is c4/(2GR2) = 2.6×10−10 J/m3 which match well with astronomical observations. We also prove that dark energy has always been ~66.7% and matter energy has been ~33.3% of the total energy and thus solve the cosmic coincidence problem. Next, we show how flatness of could be maintained since the early stage of the universe. Finally, using our model, we derive the expression for age and radius of the universe which match well with the astronomical data.

Keywords: dark energy, cosmological constant problem, cosmic coincidence problem, flatness problem, Hubble’s law, age of the universe, radius of the universe 1 Introduction nomical parameters such as dark energy (or vacuum en- ergy) responsible for accelerated expansion of the universe. This is known as cosmological constant problem. Of course, General of relativity and (or there exists some modified (Yousaf et al. quantum theory) developed in last century are thetwo 2016a,b; Yousaf 2017, 2018, 2019a; Yousaf et al. 2019b)) pillars on which the whole of modern now rests. such as f (R,T) theory (where R is Ricci scalar and T is trace Independently each of them has proved itself to be correct of energy- tensor) which can explain the gravi- with surprising accuracy. For instance, bending of tational collapse, structure formation and the expansion of coming from distant by the and precession of the universe without bringing in dark energy. But, in this perihelion of Mercury predicted by have paper, we will not follow that approach as the nature of been exactly reproduced in astronomical observations. Sim- f (R,T) is arbitrarily assumed and we are in search ilarly, prediction of value of constant by quan- of a quantum mechanical basis for astronomical observa- tum electrodynamics has been tested with a precision of tions. In other questions related to ratio of matter energy to about one part in billion. So, it is natural to expect both dark energy, flatness of space, present age and size ofuni- these jewels of physics to converge when calculating some verse, presently is unable to provide any common parameters of the universe. But, surprisingly, pre- satisfactory explanation and in this article we will attempt dictions of both these theories fall apart by unexpected to solve all these problems. Let us first discuss these prob- amounts while calculating some of the important astro- lems in detail and then progress to the proposed solutions of these in next sections. First comes the cosmological constant problem. The Corresponding Author: Biswaranjan Dikshit: Laser and original form of Einstein’s which determines Technology Division, Bhabha Atomic Research Centre, Mumbai- 400085, INDIA and Homi Bhabha National Institute, Mumbai, India; Email: [email protected]

Open Access. © 2019 B. Dikshit, published by De Gruyter. This is licensed under the Creative Commons Attribution 4.0 License B. Dikshit, Dark Energy, Cosmic Coincidence, Flatness, Age, and Size of the Universe Ë 221 the space-time structure is given by, energy. In fact, the existence of zero-point en- 8πG ergy has been experimentally demonstrated through Lamb G = T (1) µν c4 µν shift of atomic spectra, modification of magnetic moment of and (Casimir 1948; Mohideen 1998; Where Gµν is Einstein tensor, Tµν is energy-momentum ten- Lamoreaux 1997; Wilson et al. 2011) which causes an at- sor, G is Newton’s and c is speed of tractive between two closely separated conducting light. bodies. However, when we go for a quantitative comparison, When Einstein (1917) applied above equation to a static, a huge surprise appears before us. Considering all possi- isotropic and homogenous universe in the form of 3-sphere ble modes of zero-point , as shown by Weinberg containing only matter ( less dust), he found that (1989), quantum mechanical vacuum zero-point energy Eq. (1) is not satisfied. So, to solve the problem, he added a 2 π ~c density is given by ~ 4 , where lp is the minimum possi- cosmological constant Λ at the left-hand side of Eq. (1) as lp given below. ble taken to be . By putting the Λ 8πG values of constants, this quantum vacuum zero-point en- G + g = T (2) µν c2 µν c4 µν ergy density comes out to be ~ 10114 J/m3 which is about 123 where gµν is metric tensor with local space-time signature 10 times more than the value estimated from astronomi- (−,+,+,+). We have to note that the constant Λ had been ar- cal observations. This is the worst theoretical predication tificially added by Einstein just to make the universe static. in the history of which is commonly known Later on, Friedmann (1922) developed a model for an ex- as cosmological constant problem (Weinberg 1989; Banks panding universe using Einstein’s field equation given by 2004; Peebles 2003). Eq. (2) that includes the cosmological constant Λ. The Fried- In an effort to explain the astronomically observed mann’s equations are (Carroll 2001), small value of vacuum energy, few theories have been pro-

2 posed in literature which certainly are not convincing. De- (︂ R˙ )︂ 8πG Λ k = ρ + c2 − c2 (3) tails of these theories and their loop holes are described R 3 nonvac 3 R2 by Weinberg (1989). For example, assumes that corresponding to each , there exists a R¨ 4πG (︁ p )︁ Λ = − ρ + 3 + c2 (4) resulting in cancellation of energy. But how can this can- R 3 nonvac c2 3 cellation be so precise that vacuum energy density comes Where ‘R’ is the scale factor in Friedmann-Robertson- down from ~10114 J/m3 to ~5.4×10−10 J/m3? In addition, Walker (FRW) metric and physically it is the radius of till date no supersymmetric particle has been detected ex- the universe in the form of 3-sphere expanding in 4- perimentally. Another important proposal is the anthropic dimensional space. k is curvature constant which can be principle. In this principle, it is stated that even if cosmolog- +1 (for closed universe), 0 (for flat universe) or −1 (for open ical constant can take any random values, we will observe universe), ρnonvac is density of non-vacuum component it to be as it is today since it is the only value that allows (ordinary matter, and radiation). formation of and stars by gravitational conden- 8πG If we write Λ = c2 ρvac, Eq. (3) becomes, sation and creation of intelligent life who can it. (︂ )︂2 But in my view, our existence can be an effect of present R˙ 8πG k 2 = (ρnonvac + ρvac) − c (5) state of the universe, it cannot be the cause. In another ap- R 3 R2 proach to explain the present low value of vacuum energy Using the -brightness data of type Ia supernovae density, Chen and Wu (1990) have postulated cosmologi- in distant galaxies which act as standard candles, we can cal constant to be varying as R−2 where R is scale factor of measure the ratio of velocity of recession of galaxies to their universe. But, they have not given any scientific justifica- (︁ R˙ )︁ distance from us which is known as R = H (Hubble’s tion for such a variation. Similarly, many other researchers constant). Taking observed data from (Planck Collabora- (Berman 1991; Carvalho et al. 1992; Fujii and Nishioka 1990; −1 −1 tion 2016), H ≈ 67.9 km s Mpc and ρnonvac ≈ 0.31ρcrit Rubakov 2000) have also assumed relaxation of cosmo- 3H2 where critical density ρcrit = 8πG . Putting k=0 (since uni- logical constant with time, that too again without suffi- verse is found to be flat), we can calculate the vacuum en- cient reason. They have just analyzed the consequences of 2 ergy density ρvac c from Eq. (5) and its value is found to be such a variation. Understanding the origin of dark energy ~ 5.4×10−10 J/m3. This is also called dark energy density. becomes more important considering the recent work of Qualitatively, the idea of vacuum energy in general (Yousaf 2019a; Yousaf et al. 2019b) on (gravita- relativity is very well compatible with quantum field the- tional vacuum ) using modified gravity. Gravastar is a ory which considers the free space to have zero-point or 222 Ë B. Dikshit, Dark Energy, Cosmic Coincidence, Flatness, Age, and Size of the Universe

(hypothesized) celestial body that has a central vacuum is not a coincidence at all. Then, using our expressions of core containing dark energy which produces a repulsive energy density in Friedmann equations, we have explained force and thus avoids formation of singular point. how flatness of space has been maintained since the early Next unanswered question in is the prob- stage of universe. Subsequently, we mathematically prove lem of cosmic coincidence (Velten et al. 2014). Although that the Hubble’s constant is exactly equal to the reciprocal matter density decreases as 1/r3 in an expanding universe of age of universe. Thus, we estimate the present age and of radius r (from nearly ∞ to zero) and vacuum energy den- radius of universe which match well with the astronomical sity is believed to be constant, it is a surprising coincidence data. that both are of same order at the present time. Energy in the form of matter and radiation (including dark matter) is estimated to be ~32% and the rest ~68% is in the form of 2 Quantum mechanical calculation vacuum energy. Zlatev et al. (1999) had invoked the concept of an additional “tracker field” which roles down a poten- of dark energy density (Solution tial to keep ratio of dark energy to matter energy at present of cosmological constant value. However, in contrast to vacuum energy whose ef- fects have been physically observed (in form of , problem) Casimir effect etc), effect of this tracker field has never been experimentally observed. So, as correctly commented by Let us calculate the vacuum (or dark) energy density of free Velten et al. (2014), this solution seems artificial. space of the universe. As Einstein (1917) and Friedmann Next important question in is flatness prob- (1922) had visualized, we will assume the universe to be lem of space. Space is found to be flat (k=0) to the best of an isotropic, homogenous and closed 3-sphere. So, the 3-D our observational accuracy. This is possible only if the to- space of the universe is actually the surface of a 3-sphere ex- tal energy density of the space including matter/radiation isting in a four-dimensional space. All matter and radiation (︁ 3H2 )︁ becomes exactly equal to the critical density ρcrit = 8πG . are bound to this surface and the fourth along As estimated by Guth (1981), the density of universe must radial direction is not . This is due to the fact be exactly equal to the critical density with a precision of that, as per , all matter and radiation about one part in 1055 since the early stage of the universe are truly waves (Hobson 2013) and all these waves must to till date. If it were not so, the universe would have either lie on the surface of 3-sphere. Just like the vibration of a re-collapsed to a point as soon as it was born (in case of string cannot leave the string, no object (or wave packet) slightly higher density) or would have expanded at such can leave the surface of 3-sphere to move along fourth di- a large rate that galaxies and stars would not have been mension. Space itself acts as the quantum oscillator which formed by condensation (in case of slightly lower density). shows the wave behavior. So, naturally it must have the To some extent, Guth (1981, 1997) has explained this prob- zero-point (or ground state) energy. Sum of zero-point en- lem by his cosmic theory which assumes a specific ergies for all possible modes of is known as type of potential for false vacuum represented by a scalar vacuum energy. field (). But, it is not clear what caused this particu- Since, it is the quantum oscillator which is responsi- lar type of potential for the . ble for energy, we will assume that zero-point oscillator In this paper, by adopting a purely quantum mechani- for all the fundamental particles (electron, , , cal approach, we have calculated the vacuum or dark en- , etc.) is the same. Properties of particles ergy density which matches well with the astronomically other than energy appear when this oscillator in excited observed value. Thus, the earlier disagreement by a factor state couples with other fields such as Higg’s field (giving of ~10123 between quantum field theory and cosmology rise to ), (giving rise to ), disappears. Then, combining the law of etc... So, although there are 17 different fundamental parti- with this result, we have estimated the total non-vacuum cles, we will consider only a single and common zero-point (matter plus radiation) energy density of universe which is oscillation in space for all particles of same energy. again found to be nearly the same as the observed data. We It is well known that for a particle-wave trapped in have also proved that the ratio of dark energy to total energy a three-dimensional box having boundaries, three differ- and the ratio of matter energy to total energy have been of ent conditions appear i.e. each of length ‘a’, same order i.e. ~2/3 and ~1/3 respectively since the early breadth ‘b’ and height ‘c’ must be an integer multiple of stage of universe. Thus, the so called cosmic coincidence half wave length corresponding to the momentum along B. Dikshit, Dark Energy, Cosmic Coincidence, Flatness, Age, and Size of the Universe Ë 223

h h h √︁ that direction (a = nx , b = ny , c = nz ). But G~ 2px 2py 2pz where lp = c3 = Planck length in a boundary-less situation having spherical Since we have found that position uncertainty of zero- l such as Bohr’s model of atom, only one quantiza- point oscillators is always more than √p , wavelength less 2 tion condition appears i.e. circumference must be integer than this value cannot be defined and minimum possible multiple of wavelength corresponding to total momentum, lp wavelength can be taken as, λmin = √ . h 2 2πr = nλ = n p ⇒ L = pr = n~. Thus, Bohr reached at the Using the above minimum wavelength in Eq. (6) we quantized value of angular momentum of electron. Simi- get, √ larly, as the space of our universe is boundary-less (closed) 2π R 2 2π R nmax = = (10) and spherically symmetric, we will have only one quanti- λmin lp zation condition for the zero-point oscillation. So, if ‘R’ is Taking zero-point wave as mass-less (doesn’t couple with the radius of universe in the form of 3-sphere existing in Higg’s field), its velocity will be the same as the speedof four-dimensional space and ‘λ’ is wavelength of zero-point light. So, its frequency will be, ν = c/λ. Now, the total vac- oscillation, we get, uum energy of the universe due to zero-point oscillations 2π R = nλ (6) is, where n is an integer varying from 1 to nmax and the value ∑︁ 1 hc ∑︁ 1 Uvac = hν = of nmax is decided by lowest possible value of wavelength 2 2 λ λmin. Using Eq. (6), We know when the wave length is of the order of Planck nmax √︁ G −35 hc ∑︁ hc nmax(nmax + 1) length lp = ~ (=1.6×10 m), it modifies the metric of U = n = c3 vac 4πR 4πR 2 space around itself to such an extent that it becomes a 1 tiny (Saslow 1998; Lattman 2009). So, roughly Or speaking, wavelength smaller than Planck length cannot c (︂ 1 )︂ U = ~ n2 1 + (11) exist. In this paper, we will take the cutoff wavelength as vac 4R max n l max λ = √p for which a justification can be given as follows. min 2 For the zero-point oscillation which is a standing wave in Putting the value of nmax from Eq. (10) in above expression, closed universe, the momentum can be +p or −p. Thus, (︂ √ )︂2 (︂ )︂ ~c 2 2π R lp momentum uncertainty (standard deviation) is, ∆p = p. Uvac = 1 + √ 4R lp 2 2π R Then, uncertainty in position is, Or ∆x ≥ ~ (7) 2π2 cR (︂ l )︂ ~ √ p 2p Uvac = 2 1 + (12) lp 2 2π R This position uncertainty must also be more than √︁ ~G rs. Mathematically, Putting the definition of Plank length, lp = c3 , total vacuum energy becomes, ∆x ≥ rs 2 4 (︂ )︂ 2π c R lp Uvac = 1 + √ (13) Or G 2 2π R 2GE Since the total volume of space = surface area of 3-sphere ∆x ≥ c4 = 2π2R3, dividing Eq. (13) by it, we get the vacuum energy 3 (E is energy of particle) density (in J/m ) as, 1 hc pc As for zero-point oscillation, E = 2 hν = 2λ = 2 . So we Uvac uvac = get, 2π2R3 Gp ∆x ≥ (8) Or c3 4 (︂ )︂ 2 c lp Multiplying Eq. (7) and (8), uvac = ρvac c = 1 + √ (14) GR2 2 2π R 2 G~ (∆x) ≥ As R ≫ l (after early universe), second term in Eq. (13) 2c3 p and (14) becomes negligible. So, total vacuum energy and Or vacuum energy density (in J/m3) are given by, lp ∆x ≥ √ (9) 2π2c4R 2 U = (15) vac G 224 Ë B. Dikshit, Dark Energy, Cosmic Coincidence, Flatness, Age, and Size of the Universe

4 2 c pressure due to vacuum in our case becomes, u = ρ c = (16) vac vac GR2 1 2 Or mass density of vacuum is, p = − ρvac c (19) 2 c2 So, ρ = (17) vac GR2 1 pdV = − ρ c2dV Thus, from Eq. (15) and (16), we find that total vacuum 2 vac energy increases linearly with radius of universe R along Putting the value of ρvac from Eq. (14) in above and as V = 4th dimension and vacuum energy density is inversely pro- 2π2R3, portional to R2. It is interesting to note that although we had started from purely quantum mechanical approach, its 1 c4 (︂ l )︂ (︁ )︁ pdV = − 1 + √ p 2π2d R3 representative constant ~ has disappeared from the final 2 GR2 2 2π R expression of vacuum energy density in Eq. (16) migrating Or us from quantum physics (physics of small) to general rela- 2 4 (︂ )︂ tivity (physics of large). Putting the values of constants c, 3π c lp pdV = − 1 + √ (20) G and radius of universe as R=4.8×1026 m (we will prove G 2 2π R it in section 5), we get the vacuum energy density u as vac Note that work done by the system is negative implying that 5.27×10−10 J/m3 which matches well with the astronomi- its increases during expansion. cal observed value ~5.4×10−10 J/m3. Thus, we have now Removing the bracket in Eq. (13), total vacuum energy closed the earlier gap of the order of ~10123 between quan- is, tum physics and cosmology and the cosmological constant 2 4 4 problem is solved. 2π c R π c lp Uvac = + √ G 2 G Taking differential on both sides of above equation, weget, 3 Solution of Cosmic coincidence 2π2c4 dU = dR (21) problem vac G Putting Eq. (20) and (21) in Eq. (18), we get, Let the internal energy due to non-vacuum components 2 4 2 4 (︂ )︂ (ordinary matter, dark matter and radiation) of the universe 2π c 3π c lp dUnonvac = − dR + 1 + √ dR is Unonvac and the corresponding energy density is unonvac. G G 2 2π R Assuming expansion of the universe is an adiabatic process, Or first law of thermodynamics dictates, 2 4 (︂ )︂ π c 3lp dUnonvac = 1 + √ dR (22) d(Uvac + Unonvac) + pdV = 0 (18) G 2 2π R

Where p is the pressure and V is volume. Thus, we find that non-vacuum energy of the universe also Pressure due to non-vacuum component is negligible increases with radius of the universe and it is being created as speed of matter (stars and galaxies) is much less than by the expansion of space. Increase of non-vacuum energy and amount of radiation energy is less than (i.e. creation of matter and radiation) happens by way of 10−3%. So, pressure is caused by vacuum. Conventionally excitation of zero-point oscillators. Integrating Eq. (22), we in cosmology, vacuum is thought to be a perfect fluid with get, 2 negative pressure given by, p = −ρvac c . But in our case, 2 4 [︂ (︂ )︂]︂ π c 3lp R vacuum is considered to be a zero-point harmonic oscil- Unonvac = R − Ri + √ ln (23) G 2 2π Ri lator which carries its half energy in the form of and another half in the form of . Only Where Ri is initial radius of universe. Ri can be found from the potential energy part in our case can cause pressure, Eq. (10) by putting nmax equal to 1 and it is given by, Ri = l whereas kinetic part cannot as there is no change in mo- √p = 1.81 × 10−36m. Thus, Eq. (23) becomes, 2 2π mentum of zero-point oscillator (for example, in a box, a 2 4 [︂ (︂ )︂]︂ particle having kinetic energy causes pressure on wall only π c R Ri 3Ri R Unonvac = 1 − + ln (24) when it’s momentum changes by collision with wall). So, G R R Ri B. Dikshit, Dark Energy, Cosmic Coincidence, Flatness, Age, and Size of the Universe Ë 225

Dividing Eq. (24) by volume (2π2R3), we get energy density It may be noted that total energy of universe can be due to non-vacuum component given by, calculated by adding Eq. (15) and (26) which comes out to 3π2 c4 R be, U = Uvac + Unonvac = . So, in an expanding 2 Total G unonvac = ρnonvac c (25) universe, the total energy increases with R i.e. radius along 4 [︂ (︂ )︂]︂ th c Ri 3Ri R 4 dimension of space. Then, where does this energy come = 2 1 − + ln 2GR R R Ri from? This energy comes from the four-dimensional space in the form of work (PdV) (see Eq. (20)) during expansion of Just at the time of creation of universe, when R = R , i three-dimensional vacuum having negative pressure. This Eq. (24) and (25) indicate that the non-vacuum energy (mat- is opposite to the phenomena like expansion of a gas with ter/radiation) and its density were zero. The non-vacuum positive pressure in a cylinder-piston system losing energy energy density initially increased with R, became maximum in the form of work to its environment i.e. piston. Since when dunonvac = 0 and then decreased with R. So, taking the dR in our case, vacuum has negative pressure, its expansion derivative of Eq. (25) equal to zero and then numerically causes gain of energy in the form of work from its environ- solving it, we can get the radius of universe (R ) at max _density ment i.e. four-dimensional space. At the instant of creation which maximum density must have occurred. That comes of universe, Eq. (11) and (24) indicate that total energy of out to be, universe was purely dark energy (non-vacuum part was 9 lp −36 zero) and its value was ~8.7×10 J which might have been Rmax _density = 1.42 × Ri = 1.42 × √ = 2.57 × 10 m 2 2π created from . And corresponding energy density comes out to be 114 3 (unonvac)max = 9.5 × 10 /m . However, as soon as the radius of universe became 4 Solution of Flatness problem much more than initial radius or Planck length (~10−35m) i.e. R ≫ Ri, Eq. (24) and (25) reduce to As stated earlier, Einstein had artificially introduced the cosmological constant Λ in Eq. (2) just to make the uni- π2c4R Unonvac = (26) verse static. However, when the observations indicated an G expanding universe, he regretted to adding it. So, in this 4 2 c paper we will use the original version of Einstein field equa- unonvac = ρnonvac c = 2 (27) 2GR tion given by Eq. (1), but we will include the vacuum energy In the units of kg/m3, the non-vacuum mass density is, contribution in the energy-momentum tensor. So, in our case, Eq. (1) will be, c2 ρ = (28) nonvac 2GR2 8πG G = Ttotal (29) µν c4 µν Comparing Eq. (27) and (16), we find that the ratio of non- vacuum energy to the vacuum energy is ~1/2 and it is inde- Friedmann’s solution of above equation will be, −35 pendent of the radius R of universe (when it is ≫10 m). 2 (︂ R˙ )︂ 8πG k In other words, theoretically, ~ 66.7% of total energy should = ρtotal − c2 (30) R 3 R2 be in the form of dark energy and the rest 33.3% should be in the form of non-vacuum component i.e. matter (including R¨ 4πG (︁ p )︁ = − ρtotal + 3 (31) dark matter) and radiation since the beginning of universe. R 3 c2 So, cosmic coincidence problem is not really there as the total Writing ρ = ρvac + ρnonvac in the above equation and proportion of different components of energy has been a substituting quantum mechanical expressions for ρvac, constant and will remain so forever. ρnonvac and p from Eq. (17), (28) and (19) respectively, we The absolute value of predicted non-vacuum energy get, density calculated from Eq. (27) comes out to be ~2.6×10−10 3 2 J/m and astronomically observed value (Planck Collabora- (︂ R˙ )︂ 8πG (︂ c2 c2 )︂ k = + − c2 tion 2016) is ~2.4×10−10 J/m3. The reason for astronomically R 3 GR2 2GR2 R2 observed density of matter being slightly lower than the- oretical value may be due to the fact that we have some R¨ 4πG (︂ c2 c2 3 c2 )︂ inaccuracies in estimating the amount of dark matter in = − + − = 0 R 3 GR2 2GR2 2 GR2 distant galaxies. 226 Ë B. Dikshit, Dark Energy, Cosmic Coincidence, Flatness, Age, and Size of the Universe

Simplifying the above, two Friedmann equations in our 5 Estimation of age and radius of case become, 2 (︂ R˙ )︂ k = 4π − (32) universe c And Let θ be the angular distance of a from us on 3-sphere (i.e. universe). Let s and s˙ be the relative distance and rela- R˙ = Constant (33) tive velocity of recession of the galaxy from us at present time t0 when radius of universe is R. Since red shift occurs Thus, General relativity in the form of Friedmann’s solu- due to expansion of intermediate space between galaxy tion requires that Eq. (32) and (33) must be simultaneously and us and the intensity of light also depends upon present ˙ satisfied. Since Eq. (33) needs R to be constant, putting it spatial distance, measured values s and s˙ represent the rel- in Eq. (32), we get that k is also a constant in time. Thus, ative distance and relative velocity of recession of galaxy at general relativity when combined with quantum mechan- present time t . As s = Rθ and angular distance θ remains ˙ 0 ics dictates that both R and k are constants. When R = ∞, constant irrespective of size of universe, we get, energy density will be zero (from Eq. (16) and (27)) and ⃒ s˙ R˙ ⃒ so space must be flat i.e. k = 0. Since just now we have = ⃒ (35) s R ⃒ proved that k is time independent, it must be zero as soon t0 ≫ −35 Using Eq. (34) which dictates that radius of universe along as R Ri(~10 m). In other words, space must be flat or √ total density of universe must always be equal to critical the 4th dimension increases at a constant speed of 2 π c, density since the early stage of universe. Thus, flatness of we get, ⃒ √ space is achieved not by a mysterious fine tuning of density, R˙ ⃒ 2 πc 1 2 ⃒ = √ = . but due to 1/r dependence of both dark energy density and R ⃒ 2 πct t0 t0 0 matter density as given by Eq. (16) and (27). Since in the present period of measurement, t0 is approx- From Eq. (32), we get that for a flat space (k = 0), imately constant (measurements have been done during √ last century whereas present age of universe is few billion R˙ = 2 π c (34) ⃒ years), we can take R˙ ⃒ = 1 a constant which we call R ⃒ t0 √ t0 i.e. radius of universe increases at a constant speed of 2 π c “Hubble’s constant (H0)”, Thus, along the 4th dimension. This cannot be taken as violating ⃒ R˙ ⃒ 1 the special as we are here talking about ⃒ = = H0 (36) R ⃒ t0 speed of space in the form of a 3-sphere in 4-dimensional t0 space whereas limits the velocity of parti- Putting the above expression in Eq. (35), we get, ′ cles in 3-dimensional space. s˙ = H0s (Hubble s law) (37) In fact, we can prove the flatness of space irrespec- Thus, we have proved that the constant in Hubble’s law is tive of the size of universe by a simpler procedure. We exactly equal to the reciprocal of age of universe. know, if the total density becomes equal to critical den- (︁ 3H2 )︁ From Eq. (36), we can find the age of universe. Taking sity ρcrit = 8πG , then space becomes flat. Putting, H = −1 −1 √ H0 = 67.9 km s Mpc from (Planck Collaboration 2016), R˙ ˙ R and R = 2 π c, the required critical density be- age of universe comes out to be t0~14.4 billion years. This (︁ 3c2 )︁ comes, ρcrit = 2GR2 . Now total density of universe can value closely agrees with the age ~14.5 billion years esti- be calculated by adding vacuum and nonvacuum parts mated by other independent methods such as from oldest as given by Eq. (17) and (28) which comes out to be, globular clusters (Chaboyer et al. 1996). (︁ 2 )︁ 3c Present radius of universe can be calculated by putting ρtotal = ρvacuum + ρnonvacuum = 2GR2 . Thus, total density is always equal to the critical density required for flatness the value of R˙ from Eq. (34), √ of space whatever be the radius of universe. So, our model √ 2 πc R = Rt˙ 0 = 2 πct0 = (38) solves the flatness problem of universe. H0 Taking the observed value of ~67.9 km s−1 Mpc−1 for Hub- ble’s constant from (Planck Collaboration 2016), present radius of universe calculated from Eq. (38) comes out to be ~4.8×1026m (or 50 billion light years). This matches well with the approximate value ~4.4×1026 m reported by (Bars and Terning 2009). B. Dikshit, Dark Energy, Cosmic Coincidence, Flatness, Age, and Size of the Universe Ë 227

6 Conclusion References

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