Inflation and Rapid Expansion in a Variable G Model

Cosmic inflation is considered assuming a cosmologically varying Newtonian gravitational constant, G. Utilizing two specific models for, ( ) 1 G a − , where, a , is the cosmic scale parameter, we find that the Hubble parameter, H, at inception of G, may be as high as 7.56 E53 km/(s Mpc) for model A, or, 8.55 E53 km/(s Mpc) for model B, making these good candidates for inflation. The Hubble parameter is inextricably linked to G by Friedmanns’ equation, and if G did not exist prior to an inception temperature, then neither did expansion. The CBR temperatures at inception of 1 G are estimated to equal, 6.20 E21 Kelvin for model A, and 7.01 E21 for model B, somewhat lower than CBR temperatures usually associated with inflation. These temperatures would fix the size of Lemaitre universe in the vicinity of 3% of the Earths’ radius at the beginning of expansion, thus avoiding a singularity, as is the case in the ΛCDM model. In the later universe, a variable G model cannot be dismissed based on SNIa events. In fact, there is now some compelling astronomical evidence, using rise times and luminosity, which we discuss, where it could be argued that SNIa events can only be used as good standard candles if a variation in G is taken into account. Dark energy may have more to do with a weakening G with increasing cosmological time, versus an unanticipated acceleration of the universe, in the late stage of cosmic evolution.

works of Dirac in his large number hypothesis (LNH) [3] [4] [5], and Jordan [6] [7] [8] [9]. They both claimed that G must vary as a function of cosmological time, and moreover, in the case of Jordan, that G must be related to Hubble's parameter via the relation, G G H = −  . Jordan also introduced [6] a scalar field, ϕ , already in 1937, within a year of Dirac's LNH, to represent Newton's constant, realizing that G is now some sort of order parameter. The history of a variable G is long and extensive, and will not be repeated here. There have been very many theoretical and observational attempts to measure a variation in G, if it indeed exists. Some of those attempts have been presented in reference [1], and we refer the reader to that work, and references therein.
In reference [1], we sought an explanation for the cosmological constant problem. We assumed that the quintessence parameter, w, is not precisely equal to, −1, as in the ΛCDM model but rather, that its value is closer to, 0.98 w = − , as measured observationally. Within observational error, however, 1 w = − , can easily be accommodated, but perhaps this is not its exact value. Assuming that, where "a" equals the cosmic scale parameter, was much more drastic. Our theory and models went further and gave a new cosmology assuming that G does vary. In the limit where, 1 w → − , we retrieve all the standard results of the ΛCDM model. Our two models deviate appreciably from the ΛCDM model only at relatively high CBR temperatures. In this short paper, we wish to discuss some of the implications for inflation. We also wish to consider some of the ramifications in the more recent epochs, where dark energy dominates.
Inflation is a theory [10] [11] [12] [13], which is needed to explain the homogeneity, as well as the perturbations, associated the CBR temperature maps, obtained from WMAP/Planck satellite data. What is mapped occurred at photon matter decoupling, the era of last scattering, roughly 380,000 years after the Big Bang. A so-called "inflaton" field is assumed much earlier, which causes a rapid and drastic, almost explosive, expansion of the universe in its very earliest development, within 10E−32 seconds after the big bang. Within this fraction of a second, the entire universe went from roughly the size of a proton to the size of a ball, roughly 10 cm across [10]. Not all physicists are comfortable with this idea.
A-causal expansion is required where the Hubble envelop expands at faster than the speed of light. It is also a mind-boggling thought trying to imagine such a physical process, where the entire universe, as we know it, can be collapsed to what is, essentially, a singularity.
In this paper, we argue for a different interpretation, one incorporating a vari-C. Pilot International Journal of Astronomy and Astrophysics ation in G with respect to cosmological time. In reference [1], we introduced two specific models for, 1 G − , which we called models, A, and, B. Both were one parameter, non-linear functions, which mimic order parameter behavior. We believe that 1 G − is an intrinsic property of the vacuum, which involves some sort of self-organization within the vacuum, i.e., space. It is purely an artifact of space, which does not necessarily involve ordinary mass, made up of quarks and leptons. In fact, we know that the Planck mass, Pl M , and G are related by the equation, We square this result, and claim moreover, that 1 G − is an order parameter, satisfying, Here, the, In this equation, T is the temperature of the universe and 1 G − ∞ is a saturation value, achieved in the limit where, 0 T → . The constant "b" has units of temperature, but is independent of temperature. In model A, the constant, b, equals, 11.663 b = , which was determined by demanding that, 0.98 w = − . Also, Kelvin. We are also close to full saturation in the present epoch since, Saturation for all practical purposes, will occur at 10 times the current Hubble radius in this model, or when, 10 a ≅ . We are using the convention where 0 1 a = . Equations (1)(2)(3) and (1)(2)(3)(4), are modeled as a charging capacitor, and we call model A our charging capacitor model. What is charging up as a function of decreasing CBR temperature is the Planck mass squared, or the VEV, of, 2 ϕ , as seen explicitly in Equation (1-2). As mentioned, this is a very time consuming process, covering over 20 orders of magnitude, CBR temperature wise.
Model B, assumes an entirely different scaling law, namely, is the saturated value, applicable in the limit where the CBR temperature, 0 T → . We could just as well have called, The constant, "b", having units of Kelvin, has been determined to equal, 48.15 Kelvin b = , in order to guarantee that the quintessence parameter, 0.98 w = − . Again, the parameter, "b", per se, is independent of CBR temperature, even though it is measured in units of Kelvin. Here, in model B, it is found that, The nonlinear function, Equation (1)(2)(3)(4)(5), is recognized as the Langevin function, defined as, ( ) ( ) In terms of canonical dimension, magnetization has the same units as inverse mass squared, or inverse momentum squared.
Another way to rewrite Equation (1)(2)(3)(4)(5), is to remember that, ( ) where z is the redshift, and R, the Hubble radius. The temperature, 0 2.725 T = , is the current CBR temperature. We substitute "a" in place of temperature, using the above expression for, "a". In Equation (1)(2)(3)(4)(5), this gives, This order parameter surfaced [1] at a Curie temperature of, 7.01E21 Kelvin, which is very, very close to the value indicated by model A. Even though the two underlying functions, Equations (1-4) and (1-6), are entirely different and distinct functions, they lead, remarkably, to approximately the same inception temperature. The order of magnitude is perfect.
In model B, the current value for Newton's constant, 0 G , is not far from the final saturation value. In fact, we have the relation, where, C T is the Curie temperature. Then using the inception temperatures listed above, we find, The high 0 C G G values, indicated by Equations (1-8a, b), were needed to explain the cosmological constant problem, as shown in reference [1]. The cosmological constant, Λ , equals, , where ρ Λ is the mass density associated with dark energy. We see that Λ is related to both, G, and ρ Λ , and if G had a very large value in prior epochs, this would help explain the gross disparity between present, and early universe, Λ values. We can prove, namely, that [1], This still does not take into account dark matter, nor dark energy, but it does factor in what is definitely known. If we multiply Equation (2)(3)(4)(5)(6)(7)(8)(9), by this correction factor, * 106.75 g = , we will have effectively taken into account, as well, the radiation that coalesced as material particles between then and now.
What remains is to find C a . But this is easy since we know the temperatures at inception, and, For models A, and B, we find, respectively, that These values are very close to one other because the inception temperatures were nearly equal. We next substitute these cosmic scale factors into Equation We do this next. We substitute Equations (1-11a, b), and (1-8a, b), into Equa-  1-10a, b), that This is roughly 3 percent of the earth's radius. This would be our initial estimate for the size of Lemaitre's cosmic egg at the beginning of expansion.

Dark Energy and Subsequent Late Epoch Development
We now turn to the later stages of cosmic evolution where we have dark energy. We want to consider, specifically, the increased luminosity distance associated with SNIa events, which led one to conclude that the universe is currently expanding more rapidly, than anticipated. Consider a specific SNIa event, where the energy flux, measured on earth bound satellites, fixes a specific distance to the source using the luminosity-distance-flux relation. SNIa events, i.e., explosions, make for good standard candles because of their brightness, and excellent predictable luminosity. The observed distances, measured in the late 1990's, and after, suggest that the universe is expanding faster than we thought. We have an unanticipated acceleration leading one to surmise, erroneously we believe, that in the later stages of cosmic evolution, a type of antigravity or negative pressure surfaces, which we identify as dark energy. What happens however, if G varies cosmologically?
The luminosity of a SNIa event varies as [27], and if G is unequal to a constant, the luminosity would vary. Since 1 G − increases as cosmological time increases, 3 2 G − must also increase. So gravity gets weaker as time progresses, and the universe would appear to accelerate more expansion wise. The two functions that we introduced for 1 G − in reference [1] can be used to model this behavior.
Taking this a step further, we know that the energy flux received by an observer here on earth is related to the luminosity distance by the relation, In this equation, L d ′ is the perceived and un-anticipated, accelerated luminosity distance, leading to the notion of dark energy. By contrast, L d , is the true luminosity distance. The increase in L d ′ over L d is really due to a decrease in 1 G − if one goes back in time. Equation, (2-1), can namely be rewritten as ( ) It is apparent by this equation, that if G increases, then, L d ′ also increases.
Depending on the look-back time, at a specific redshift, the G is stronger in value, and thus by Equation  (2)(3)(4)(5) And therefore, by Equation (2-2), we expect, We emphasize that both functions, Equation (1)(2)(3)(4), and Equation (1-6), for Let us now carry out the same analysis for one further redshift. We now consider a redshift, 2 3 z = , or, There is some astronomical evidence that higher redshifted SNIa luminosities are weaker. Thorsett [24], analyzed the energy release in SNIa explosions both near, at low z, and far, at higher z. The lookback times were between 1 Gyr, and,

Summary and Conclusion
Cosmic inflation, i.e., the rapid expansion of the universe in its earliest phase, has been considered assuming a cosmologically varying G. The Friedmann equation, in particular, is invoked to argue that Hubble expansion is only possible if G is unequal to zero. If G varies with cosmological time, then it must be an intrinsic property of the vacuum. Moreover, it must be related to a scalar field, ϕ , as first suggested by Jordan. The Planck mass squared,  1-12a, b), and (1-13a, b). We believe that these expansion rates could be interpreted as inflation, where the "inflaton" field of inflation is replaced by Jordan's scalar field, ϕ , above. Beyond this point, temperature-wise, G, simply did not exist, and there was no expansion of space, as determined by the Friedmann equation. It is hypothesized that when the universe started to expand, it had a finite size, of the order of the earth's radius. In the ΛCDM model, expansion started from a singularity.
Finally, a variable G model cannot be dismissed based on SNIa events. In fact, there is now some very solid observational/astronomical evidence indicating otherwise. A variable G may be needed to interpret SNIa energy release properly, as well as rise and fall times for these events. See Equations (2-1) and (2-2). Specific predictions for luminosity distance are possible using our two models for, . And for a larger redshift, 2 3 z = , we obtain Equations (2-7) and (2)(3)(4)(5)(6)(7)(8). It may well turn out that SNIa events, in order to serve as good standard candles, need to include a variation in Newton's constant, G. Without taking into account a weakening G with an increase in cosmological time, erroneous results and interpretations occur. Dark energy may have more to do with a weakening in G with decreased redshift, than anything else.