A resolution of the Trans-Planckian problem in the R_h=ct universe
aa r X i v : . [ a s t r o - ph . C O ] S e p Received 20 May 2020; Revised 17 August 2020; AcceptedDOI: xxx/xxxx
COMMENT
A resolution of the Trans-Planckian problem in the 𝑅 h = 𝑐𝑡 universe Fulvio Melia* Departments of Physics and Astronomy, andthe Applied Math Program, The Universityof Arizona, Tucson, Arizona, U.S.A.
Correspondence *Email: [email protected]
The recent measurement of a cutoff 𝑘 min in the fluctuation power spectrum 𝑃 ( 𝑘 ) ofthe cosmic microwave background may vitiate the possibility that slow-roll inflationcan simultaneously solve the horizon problem and account for the formation of struc-ture via the growth of quantum fluctuations in the inflaton field. Instead, we showthat 𝑘 min may be interpreted more successfully in the 𝑅 h = 𝑐𝑡 cosmology, as the firstmode exiting from the Planck scale into the semi-classical Universe shortly after theBig Bang. In so doing, we demonstrate that such a scenario completely avoids thewell-known trans-Planckian problem plaguing standard inflationary cosmology. KEYWORDS: cosmological parameters; cosmology: observations; cosmology: early Universe; cosmology: theory;gravitation; cosmology: inflation
The Friedmann-Lemaître-Robertson-Walker (FLRW) metric,based on the cosmological principle and its assumption ofisotropy and homogeneity on large scales, is the backbone ofmodern cosmology. All the available observational evidenceappears to support its essential spacetime basis (Melia, 2020),so any conceptual or foundational hurdles arising with theexpansion of the Universe are attributed to other factors—notably an incomplete understanding of the physics underlyingthe evolution of its contents.Over the past four decades, several crucial amendments andadditions have been introduced to the basic picture in order toaddress some of these difficulties, chief among them the well-known horizon problem associated with the uniformity of thecosmic microwave background (CMB) temperature, 𝑇 cmb . Inthe context of Λ CDM, CMB photons emitted near the surfaceof last scattering (LSS) at redshift 𝑧 ∼ 1080 from oppositesides of the sky would be causally disconnected without ananomalous accelerated expansion in the early Universe (Melia,2013). Yet 𝑇 cmb has the same value in all directions, save for ∼ 10 −5 variations associated with fluctuations seeded at, orshortly after, the Big Bang. A very elegant solution to this problem was introduced inthe early 1980’s (Guth, 1981), based on an expected phasetransition in grand unified theories (GUTs), when the strongand electroweak forces may have separated at an energy scale ∼ 10 GeV, or ∼ 10 −35 seconds after the Big Bang. As long asthe scalar field, 𝜙 , associated with this spontaneous symmetrybreaking had the ‘right’ potential, 𝑉 ( 𝜙 ) , one could envisagean evolution at almost constant energy density, 𝜌 𝜙 , produc-ing a transient near-de Sitter cosmic expansion (Linde, 1982).Such an inflationary phase would have exponentially stretchedall observable features well beyond the Hubble radius, 𝑅 h = 𝑐 ∕ 𝐻 , where 𝐻 ( 𝑧 ) is the redshift-dependent Hubble parame-ter, causally connecting the spacetime throughout the visibleUniverse today.Perhaps even more importantly, this event is believedto have also produced the large-scale structure viathe seeding of quantum fluctuations in 𝜙 and theirsubsequent growth to classically relevant scales dur-ing the inflated expansion (Kodama & Sasaki, 1984;V. F. Mukhanov, Feldman, & Brandenberger, 1992). A nearscale-free spectrum 𝑃 ( 𝑘 ) would have been generated as modeswith comoving wavenumber 𝑘 successively crossed 𝑅 h andclassicalized, freezing their amplitude at a mode-dependentcrossing time 𝑡 𝑘 (Maldacena, 2003). Thus, inflation appears Fulvio Melia to have simultaneously solved the 𝑇 cmb horizon problem andprovided an explanation for the origin of 𝑃 ( 𝑘 ) .In spite of this initial success, however, the inflationaryparadigm is nonetheless conceptually incomplete for severalreasons. For example, the recent discovery of the Higgs parti-cle (Aad et al., 2012) has reminded us that Λ CDM is subjectto several horizon problems—at several different epochs—notjust one, so the GUT transition at ∼ 10 −35 seconds is look-ing more like an overly customized solution focusing solelyon 𝑇 cmb , rather than providing an over-arching paradigm toaccount for our entire causally-connected Universe. A secondwell-motivated phase transition should have occurred whenthe electric and weak forces separated at a critical temperature 𝑇 Higgs ∼ 159 . . GeV, i.e., 𝑡 ∼ 10 −11 seconds after theBig Bang (Barr, Dolan, Englert, de Lima, & Spannowsky,2015; Dolan, Englert, & Spannowsky, 2012;Fileviez Pérez, Patel, Ramsey-Musolf, & Wang, 2009;Noble & Perelstein, 2008; Weir, 2018), too far beyond theGUT scale to have been affected by the hypothesized firsttransition (Melia, 2018c). This second spontaneous symmetrybreaking would have inevitably led to its own horizon problem,having to do with the ‘turning on’ of the Higgs field and itsvacuum expectation value, which today appears to be univer-sal, even on scales exceeding the regions that were causallyconnected at the time of the electroweak phase transition.This is not so much an argument against inflation per se,though it does weaken the claim that a GUT phase transitioncould account for all of the major features we see today; itapparently does not. A more serious problem with the slow-rollinflationary paradigm has been uncovered by another recentstudy of the angular correlation function measured in the CMBby Planck (Planck Collaboration et al., 2018).The inflationary model faces several other hurdles, notmerely possible inconsistencies with the horizon problems.For example, recent observations of the CMB anisotropieshave apparently eliminated a broad range of possible infla-tionary scenarios, disfavoring all the simplest potentials(Ijjas, Steinhardt, & Loeb, 2013). And the remaining range ofinflaton fields are subject to serious drawbacks, making it muchmore difficult to understand how the universe acquired itsinitial conditions. Along with this outcome, many of the best-motivated inflationary scenarios appear to have been ruledout, as we shall see below. According to this recent anal-ysis (Liu & Melia, 2020; Melia & López-Corredoira, 2018),none of the slow-roll potentials based on the ‘Quadratic,’‘Higgs-like’ and ‘Natural’ forms can account for the observedpower spectrum while simultaneously mitigating the horizonproblem. Having said this, it is not really our goal in this
Note to argue against the inflationary paradigm, though the diffi-culties faced by inflationary cosmology certainly add to our motivation for pursuing the alternative interpretation describedhere.In the standard picture, the solution to the 𝑇 cmb horizonproblem, and a generation of a near scale-free fluctuationspectrum 𝑃 ( 𝑘 ) , are intimately connected via the initiationand extent of the inflationary phase. But the CMB angular-correlation function now provides compelling evidence—ata confidence level exceeding 𝜎 —that 𝑃 ( 𝑘 ) has a non-zerocutoff 𝑘 min = (4 .
34 ± 0 . 𝑟 cmb , where 𝑟 cmb is the comov-ing distance to the LSS (Melia & López-Corredoira, 2018).Since 𝑘 min would have been the first mode crossing 𝑅 h dur-ing inflation, it would signal the precise time, as a function of 𝑉 ( 𝜙 ) , at which the de Sitter expansion started. Unfortunately,its measured value shows that none of the slow-roll potentialsproposed thus far can simultaneously account for the unifor-mity of 𝑇 cmb across the sky and the observed 𝑃 ( 𝑘 ) in the CMB(Liu & Melia, 2020). The conclusion from this is that, if slow-roll inflation is to work, it must function in a more complicatedway than has been imagined thus far.As we shall see in this Note , the measurement of 𝑘 min in the angular correlation function of the CMB not onlyconstrains the time when inflation could have started, butapparently provides direct evidence of quantum fluctuationsat the Planck scale. This topic broaches one of the mostserious fundamental problems with inflation, one that haseluded satisfactory resolution for over three decades. It isgenerally understood that to solve the horizon problem in Λ CDM, a minimum of 60 e-folds of inflationary expan-sion must have occurred, even more in many variants of thebasic model. Thus, cosmological scales of observational rel-evance today must have expanded from sub-Planckian wave-lengths at the start of inflation (R. H. Brandenberger & Martin,2001; Easther, Greene, Kinney, & Shiu, 2001; Kempf, 2001;Niemeyer, 2001). But the physics we have today cannot ade-quately handle such processes, a situation known as the ‘trans-Planckian problem’ (TP) (Martin & Brandenberger, 2001).(See R. H. Brandenberger & Martin (2013) for a detailedreview.) This signals a potentially fatal incompleteness ofinflationary theory at a fundamental physics level.The key here is how one should interpret the evolutionof quantum fluctuations seeded well below the Planck scale,where quantum mechanics and general relativity as we knowthem today are probably not valid (Niemeyer & Parentani,2001). The problems that derive from attempting to followthis trans-Planckian evolution have been well documented. Forexample, attempts to renormalize a scalar theory with nontriv-ial initial conditions in flat space run into possible divergencesconfined to the surface where the initial conditions are imposed(Collins & Holman, 2005).As we shall see in the next section, the development of aquantum theory of gravity could resolve such issues, allowing ulvio Melia us to follow the growth and evolution of quantum fluctua-tions from birth to their exit into the semi-classical universe(beyond the Planck domain). Attempts have also been madeto modify the dispersion relation for quantum modes on shortscales (see, e.g., refs. Ashoorioon, Chialva, & Danielsson2011; Zhu, Wang, Kirsten, Cleaver, & Sheng 2016), or to alterthe Heisenberg uncertainty relation (Easther et al., 2001), inorder to uncover possible observational signatures that onemay detect in the CMB, allowing us to make some progress inidentifying new physics at the Planck scale.Our goal in this Note is to provide an alternative interpre-tation of the cutoff 𝑘 min measured in the primordial powerspectrum 𝑃 ( 𝑘 ) , in order to mitigate this current need of hav-ing to follow the evolution of inflationary quantum fluctuationswhere our semi-classical theories are unlikely to be valid. Λ CDM
One can easily understand why our inability to analyze physicsbelow the Planck scale constitutes a potentially insurmount-able problem. The Planck mass is a unit of mass definedsolely using fundamental and universal units, and is given by 𝑀 P ≡ √ ℏ𝑐 ∕ 𝐺 . As it turns out, the Planck length associatedwith 𝑀 P is of the same order as the Schwarzshild radius andthe Compton wavelength of the Planck mass, suggesting thatit represents a transition from classical general relativity tothe quantum gravity domain. Given this ‘physical’ interpreta-tion, we shall use in this Note a slightly different value of thePlanck mass, which we shall call 𝑚 P = √ 𝜋ℏ𝑐 ∕ 𝐺 (see below),derived from strictly setting its Compton wavelength equal toits Schwarzschild radius.It is not difficult to see that, since the former length scaleincreases as the latter shrinks towards the Big Bang, it is sim-ply not possible to characterize the behaviour of modes belowthe Planck scale using quantum mechanics and general relativ-ity separately. The semi-classical physics we use to describethe evolution of quantum fluctuations as the Universe expandsdoes not apply for mode scales shorter than their Comptonwavelength (R. H. Brandenberger & Martin, 2013).This problem manifests itself in several ways, particularlyvia the mode normalization that one must use to calculate 𝑃 ( 𝑘 ) for a comparison with the CMB data. The amplitudeof the modes is typically inferred by minimizing the expec-tation value of the Hamiltonian, but with a time-dependentspacetime curvature at the Planck scale, the frequencies them-selves depend on time and non-inertial effects. Early attemptsat addressing this issue extended the birth of fluctuation modesinto the very distant conformal past, well below the Planckscale, arguing that the simple harmonic oscillator is recovered there, allowing one to impose a Minkowski vacuum—calledthe Bunch-Davies vacuum in this context (Bunch & Davies,1978)—as the background for the fluctuations. But given thatthe physics below the Planck scale is unknown, we have aconceptual problem understanding whether or not the Bunch-Davies vacuum is even the correct choice for sub-Planckianmodes.We should point out, however, that this trans-Planckianproblem arises primarily from the use of quantum mechanicswith (the classical theory of) general relativity. It is believedthat this problem may disappear with a viable theory of quan-tum gravity, such as Wheeler-DeWitt theory, loop quantumgravity or string theory (Kiefer, 2004). Significant effort isbeing expended in studying such models, which are establishedby using quantization techniques on symmetry-reduced gen-eral relativity. For a large class of wave functions, the homo-geneous and isotropic FLRW metric even loses its singularity(Ashtekar, Corichi, & Singh, 2008).But such an approach is not entirely problem free either.These models suffer from the long-standing measurementproblem in quantum mechanics, having to do with the ambi-guity of when exactly collapses happen and how a unique,determanistic outcome is selected. The measurement problemis even more severe in cosmology, because the universelacks outside observers or measurement devices that couldhave collapsed the wave function. There is also the problemof time, given that the wave function is static in both theWheeler-DeWitt theory and loop quantum gravity (Kiefer,2004; Kuchař, 2011), making it difficult to envisage temporalevolution, e.g., to ascertain whether the universe is expand-ing or contracting. Finally, such quantum gravity theories arebased solely on a wave function, without an actual metric, soit is difficult to determine the behavior of quantum fluctuationsin the remote conformal past, where singularities might haveemerged anyway (Dewitt, 1967).The consensus today is that Planck-scale physics prob-ably should have created an imprint on the CMB, butwith no established theory of quantum gravity, no oneknows how to predict such features with any confi-dence. Instead, the approaches taken over the past twodecades have been based on phenomenological treatments,including (1) modifications to the dispersion relation forquantum modes on short scales (Ashoorioon et al., 2011;R. H. Brandenberger & Martin, 2001; Jorás & Marozzi, 2009;Martin & Brandenberger, 2001; Niemeyer, 2001; Zhu et al.,2016); (2) the use of string-inspired changes to the Heisenberguncertainty relation (Easther et al., 2001; Hassan & Sloth,2003; Kempf, 2001); and (3) noncommutative geometry(R. Brandenberger & Ho, 2002; Chu, Greene, & Shiu, 2001;Lizzi, Mangano, Miele, & Peloso, 2002). Fulvio Melia
All of these are really probes of the CMB to sug-gest how basic theory ought to be modified rather thanrobust attempts at using a well-justified model of physicsat short distances to predict a trans-Planckian signature.To understand the scale we are considering here, wedefine the Planck length 𝜆 P to be the Compton wavelength 𝜆 C ≡ 𝜋 ∕ 𝑚 P of a (Planck) mass 𝑚 P for which 𝜆 C equalsits Schwarzschild radius 𝑅 h ≡ 𝐺𝑚 P . The Planck energyis therefore 𝑚 P ≈ 1 .
22 × 10 GeV. Estimates of how bigtrans-Planckian corrections might be, based on the abovephenomenological approaches, range from ( 𝜆 P ∕ 𝑅 ℎ [ 𝑡 inf ]) (see, e.g., Kaloper, Kleban, Lawrence, & Shenker 2002;Kempf & Niemeyer 2001) to as large as 𝜆 P ∕ 𝑅 ℎ ( 𝑡 inf ) (R. H. Brandenberger & Martin, 2002; Danielsson, 2002;Easther et al., 2001; Easther, Greene, Kinney, & Shiu, 2002).In these expressions, 𝑅 h ( 𝑡 inf ) is the Hubble radius duringinflation (which is more or less constant in the slow-rollapproximation).Thus, if inflation is associated with a GUT phase transitionat ∼ 10 GeV, these phenomenologically motivated correc-tions fall in the range −6 to −3 . Additional support for sucha claim—especially towards the high-end of this range—isprovided by arguments (R. H. Brandenberger & Martin, 2002;Danielsson, 2002) that curvature effects at the Planck scaleprobably produce deviations of the 𝜙 quantum state from thelocal vacuum state on the order of 𝜆 P ∕ 𝑅 ℎ ( 𝑡 inf ) , but no onereally knows for sure. If reasonable, this range includes effectspotentially large enough to affect the primordial power spec-trum 𝑃 ( 𝑘 ) in measurable ways (see, e.g., Easther et al. 2002).Of course, a final resolution of whether or not trans-Planckianeffects manifest themselves observationally must await theformulation of a well-motivated quantum gravity theory.Certainly, the measurement of 𝑘 min already seems to argueagainst the premise that inflation might have started earlyenough to solve the temperature horizon problem, whilesimultaneously explaining the origin of 𝑃 ( 𝑘 ) . All threemajor satellite missions designed to study the CMB—COBE(Hinshaw et al., 1996); WMAP (Bennett et al., 2003); and Planck (Planck Collaboration et al., 2018)—have uncoveredseveral large-angle anomalies with the temperature fluctua-tions, including missing correlations at angles exceeding ∼ 60 ◦ and an unexpectedly low power in multipoles ≲ 𝓁 ≲ .These unexpected features stand in sharp contrast to the gen-eral level of success interpreting the CMB anisotropies with Λ CDM at angles ≲ ◦ , i.e. at 𝓁 ≳ .After a careful re-analysis of the latest release of the Planck data, it now appears that both of these discrepancies have com-mon origin: a sharp cutoff of the primordical power spectrum 𝑃 ( 𝑘 ) at the aforementioned minimum wavenumber 𝑘 min . In thefirst of these studies (Melia & López-Corredoira, 2018), such a cutoff was shown, not only to suppress the expected corre-lation at large angles but, to actually significantly improve thefit of the angular correlation function at all angles. This analy-sis demonstrated that a zero 𝑘 min was ruled out at a confidencelevel exceeding 𝜎 . The second study focused on the impactof 𝑘 min on the CMB angular power spectrum itself, and con-cluded that the missing power at low multipoles was equallywell explained by this cutoff, while none of the other 11 or12 Λ CDM parameters optimized by
Planck were materiallyaffected by the introduction of 𝑘 min .The combined result of these two analyses is that a singlevalue of 𝑘 min completely mitigates both large-angle anoma-lies. But as shown in Liu & Melia (2020), the interpretationof this cutoff as the first mode to cross the Hubble radius 𝑅 h once slow-roll inflation began is then inconsistent with theaccelerated expansion required to provide us with a causally-connected Universe today. The principal reason is that the timeassociated with this crossing was too late to have permittedthe quasi-de Sitter phase to have inflated the universe suffi-ciently to solve the CMB temperature horizon problem. In thenext section, we present an alternative interpretation of 𝑘 min that avoids these conceptual problems and, at the same time,completely eliminates the trans-Planckian inconsistency. 𝑅 H = 𝐶𝑇 The FLRW cosmology known as the 𝑅 h = 𝑐𝑡 universe(Melia, 2007, 2016, 2017b; Melia, 2020; Melia & Abdelqader,2009; Melia & Shevchuk, 2012) is essentially Λ CDM, thoughwith an additional constraint motivated by both the observa-tional evidence and a careful application of the Local FlatnessTheorem in general relativity. It too has an energy density 𝜌 dominated by various combinations of matter, 𝜌 m , radia-tion, 𝜌 r and dark energy, 𝜌 de , depending on the cosmic epoch,but the equation-of-state of this ‘cosmic fluid’ always satisfiesthe so-called zero active mass condition, 𝜌 + 3 𝑝 = 0 , where 𝜌 = 𝜌 m + 𝜌 r + 𝜌 de and 𝑝 is the corresponding total pressure 𝑝 = 𝑝 m + 𝑝 r + 𝑝 de . An introductory review of this model may befound in ref. Melia (2019b), and a more thorough presentationof its foundational support, both observational and theoretical,will be presented in the upcoming monograph Melia (2020),to be released in Fall, 2020.By now, this model has been subjected to comparative testswith basic Λ CDM using over 27 different kinds of cosmologi-cal data, and has accounted for the observations at least as wellas the standard model, actually better than latter in the major-ity of cases. A recent summary of these comparative tests maybe found in Table 2 of Melia (2018b). ulvio Melia The 𝑅 h = 𝑐𝑡 cosmology was first postulated on the basisof such empirical evidence showing that the apparent horizon 𝑅 h ≡ 𝑐 ∕ 𝐻 , averaged over a Hubble time, equals the comov-ing distance, 𝑐𝑡 , light could have traveled since the Big Bang(Melia, 2018b). In general relativity, this horizon separates nullgeodesics receding from the observer from those approachinghim/her, but is generally not the same as the event horizon,which signals the causal limit in the asymptotic future (Melia,2018a). As such, there is no reason why 𝑅 h should alwaysequal 𝑐𝑡 , unless there exists some specific, fundamental reasonforcing this condition.Recently, the theoretical basis for 𝑅 h = 𝑐𝑡 was strength-ened considerably with a thorough re-examination of the lapsefunction (i.e., the coefficient 𝑔 𝑡𝑡 in the FLRW metric; Melia2019a. The issue of whether or not a lapse function 𝑔 𝑡𝑡 = 1 isconsistent with a non-inertial Hubble flow has been paid scantattention in the past. In this recent work, the Local FlatnessTheorem (Weinberg, 1972) in general relativity was used toprove that 𝑔 𝑡𝑡 = 1 is in fact valid for only two specific equationsof state: an empty Universe with 𝜌 = 𝑝 = 0 (i.e., Minkowskispace) and the aforementioned zero active mass condition, 𝜌 + 3 𝑝 = 0 . As we now know, a pressure 𝑝 = − 𝜌 ∕3 pro-duces a constant expansion rate and, more importantly, forcesthe equality 𝑅 h = 𝑐𝑡 . The empirical evidence pointing to thisconstraint was therefore a pre-confirmation of the subsequenttheoretical analysis based on the Local FLatness Theorem,showing that the use of FLRW is valid only when the cosmicfluid satisfies this equation of state.A notable feature of the expansion implied by this scenariois that it lacks any horizon problem, eliminating the need foran inflated expansion of the early Universe. Thus, if the zeroactive mass condition was evident at the earliest times, 𝑡 , itis straightforward to show (Melia, 2017a) that an incipient(though non-inflationary) scalar field 𝜙 would have had thewell-defined potential 𝑉 ( 𝜙 ) = 𝑉 exp { − 2 √ 𝜋𝑚 P 𝜙 } . (1) 𝜙 is therefore a special member of the class of minimally cou-pled fields explored in the 1980’s, that produced power-lawinflation (Abbott & Wise, 1984; Barrow, 1987; Liddle, 1989;Lucchin & Matarrese, 1985) except that, with the zero activemass equation-of-state, this 𝜙 produced a constant expansionrate 𝑎 ( 𝑡 ) = 𝑡 ∕ 𝑡 and did not inflate.In 𝑅 h = 𝑐𝑡 , quantum fluctuations in 𝜙 with a wavelength 𝜆 𝑘 < 𝜋𝑅 h , where 𝑘 is the comoving wavenumber and 𝑅 h is the Hubble radius, oscillate, while those with 𝜆 𝑘 > 𝜋𝑅 h do not (Melia, 2017a). Thus, mode 𝑘 oscillated in the semi-classical Universe once it emerged across the Planck scale.But the critical question is “When did it emerge?" From theexpression 𝑘 = 2 𝜋𝑎 ( 𝑡 )∕ 𝜆 𝑘 ( 𝑡 ) , it is clear that the observed value of 𝑘 min indicates the time 𝑡 min when the first mode appeared.Therefore, 𝑡 min = 4 . 𝑡 P ln(1 + 𝑧 cmb ) , (2)in terms of the redshift, 𝑧 cmb , at the surface of last scattering.In the concordance Λ CDM model, 𝑧 cmb ∼ 1080 , for which 𝑡 min ∼ 0 . 𝑡 P . With the expansion scenario implied by 𝑅 h = 𝑐𝑡 ,this redshift could be quite different, but the dependence of 𝑡 min on the location of the last scattering surface is so weak, thateven a redshift 𝑧 cmb ∼ 50 would result in an initial emergencetime of 𝑡 min ∼ 1 . 𝑡 P . Therefore, it appears that 𝑘 min in 𝑅 h = 𝑐𝑡 represents the first mode exiting the Planck region at about thePlanck time, a compelling indication that the cutoff 𝑘 min cor-responds to the first mode that could have physically emergedinto the semi-classical Universe after the Big Bang.Unlike the situation with an inflaton field, in which thesemodes were seeded in the Bunch-Davies vacuum and oscil-lated across the trans-Planckian region, the quantum fluctu-ations associated with a non-inflationary scalar field in the 𝑅 h = 𝑐𝑡 cosmology could well have been formed at the Planckscale and then evolved according to standard physical princi-ples in the semi-classical Universe. Such an idea—that modescould have been created at a particular (perhaps even fixed)spatial scale—is not new. It has been proposed and discussed inthe literature before, notably by Hollands & Wald (2002), butalso by R. Brandenberger & Ho (2002) and Hassan & Sloth(2003), among others.Whereas inflaton quantum fluctuations would have beenborn randomnly at various times in the distant conformal past,our new proposal is that all of the non-inflaton fluctuationsemerged into the semi-classical universe at the same spatialscale , i.e., the Planck length. They could very well have hada past history prior to reaching 𝜆 P , but the difference is thatwe don’t need to know how to dynamically evolve them belowthis scale using our current version of quantum mechanics andgeneral relativity in order to self-consistently evolve them at 𝑡 ≥ 𝑡 𝑘 . In contrast, the conventional picture requires that weuse these semi-classical theories in a domain where they areprobably not valid. Without invoking a Bunch-Davies vacuumand following their evolution prior to 𝑡 P , we cannot ensure thatthe inflaton field simultaneously fixed the horizon problem andproduced the required primordial power spectrum 𝑃 ( 𝑘 ) .Another key difference between the quantum fluctuationsassociated with an inflaton field and those generated withthe field potential in Equation (1), is that the solutionto the Mukhanov-Sasaki equation (Kodama & Sasaki, 1984;V. Mukhanov, 2005), using an expansion factor 𝑎 ( 𝑡 ) = 𝑡 ∕ 𝑡 consistent with this potential, can be easily shown to have a constant frequency—dependent only on the time 𝑡 𝑘 at whichthe quantum fluctuation emerged out of the Planck domain.It was therefore a true harmonic oscillator and its amplitude Fulvio Melia may be determined via canonical quantization in flat space-time or, equivalently, by minimizing the Hamiltonian. This isdue entirely to the zero active mass condition, which producedzero acceleration, even at the Planck scale. Compare this withthe conventional case, in which the extreme spacetime curva-ture at 𝜆 P obviates such a straightforward approach and onemust instead evolve the quantum fluctuations starting with theBunch-Davies vacuum in the very remote past.Assuming that all subsequent modes continued to emergeacross the Planck scale with a wavelength 𝜆 𝑘 = 2 𝜋𝜆 P ,though at progressively later times 𝑡 𝑘 ≡ 𝑘𝜆 P 𝑡 , it is triv-ial to show (Melia, 2017a) that the resultant power spec-trum is almost scale free, with an index 𝑛 𝑠 slightly lessthan one, consistent with the value measured by Planck (Planck Collaboration et al., 2018). Thus, a non-inflationaryscalar field in the 𝑅 h = 𝑐𝑡 universe can account for both themeasured cutoff 𝑘 min and for the observed distribution of fluc-tuations in the CMB. Most importantly for the main theme ofthis paper, the first reliable measurement of a minimum cutoffin the power spectrum 𝑃 ( 𝑘 ) signals a direct link between theCMB anisotropies—and the subsequent formation of structurein the Universe—and quantum fluctuations at the Planck scale.In so doing, this interpretation eliminates one of the princi-pal inconsistencies with the basic slow-roll inflationary model,i.e., the well-known trans-Planckian problem.In short, this principal distinction between the standardinflationary scenario and the mechanism described here isthat quantum fluctuations in the former had to be seeded onscales well below the Planck length, at a time well beforethe Planck time in order to produce the observed anisotropiesin the CMB, while in the latter they could have formed atthe Planck scale, with the first emerging at the Planck time.All subsequent modes would also have formed at the Planckscale, though after the Planck time, thereby producing a nearscale free spectrum, and always evolving dynamically accord-ing to standard quantum mechanics and general relativity inthe semi-classical Universe. The mechanism we are describinghere therefore completely avoids the trans-Planckian problem,because our treatment of the quantum fluctuations relies solelyon the physics we know, based on initial conditions at thePlanck scale—not below it. In this paper, we have discussed the implications of the factthat, in addition to the well-studied power spectral index 𝑛 𝑠 and amplitude of the CMB fluctuations, we now have arobust measurement of a third parameter characterizing theprimordial perturbation spectrum, i.e., the wavenumber cutoff 𝑘 min , which differs from zero at a confidence level exceed-ing 𝜎 (Melia & López-Corredoira, 2018). This cutoff appearsto invalidate basic slow-roll inflationary models attemptingto simultaneously account for the 60 e-folds of exponentialexpansion at the GUT scale and the generation of anisotropiesin the CMB from quantum fluctuations in the inflaton field(Liu & Melia, 2020). An additional well-known inconsistencywith this scenario is the trans-Planckian problem, referringto the required transition of modes from below the Planckscale into the semi-classical Universe, a process that cannotadequately be described by quantum mechanics and generalrelativity separately.Contrasting with this deficiency in the standard model, wehave also demonstrated that the interpretation of 𝑘 min in the 𝑅 h = 𝑐𝑡 cosmology suggests it corresponds to the first quan-tum fluctuation that could have physically emerged from thePlanck scale shortly after the Big Bang. This scenario thusavoids the trans-Planckian problem if one invokes the idea thatall fluctuations in the incipient scalar (though non-inflationary)field were seeded at a fixed spatial scale—in this case, thePlanck scale—though at progressively later times dependingon the wavenumber 𝑘 of the mode. This interpretation is fullyconsistent with the quantum mechanical meaning of the Plancklength, representing the shortest physical size of any causallyconnected region in the early Universe.Looking to the future, this interpretation of 𝑘 min may offerclues concerning how to extend our current semi-classicaldescription of the early Universe to scales below the Plancklength, thereby heralding the initiation of an observationally-motivated quantum gravity theory. In concert with such ideas,we point out that, if the 𝑅 h = 𝑐𝑡 cosmology turns out to be cor-rect, the potential of the (non-inflationary) scalar field presentjust after the Big Bang is precisely known (Eq. 1), allowing useventually to also focus more directly on possible extensionsto the standard model of particle physics. ACKNOWLEDGMENTS
I am grateful to the anonymous referee for his/her commentsand suggestions, which have improved the presentation in thispaper. I also acknowledge Amherst College for its supportthrough a John Woodruff Simpson Fellowship.
Author contributions
Fulvio Melia is the sole author of this paper.
Financial disclosure
None reported. ulvio Melia Conflict of interest
The authors declare no potential conflict of interests.
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