Large scale density perturbations from a uniform distribution by wave transport

It has long been known that a uniform distribution of matter cannot produce a Poisson distribution of density fluctuations on very large scales $1/k > ct$ by the motion of discrete particles over timescale $t$. The constraint is part of what is sometimes referred to as the Zel'dovich bound. We investigate in this paper the transport of energy by the propagation of waves emanating {\it incoherently} from a regular and infinite lattice of oscillators, each having the same finite amount of energy reserve initially. The model we employ does not involve the expansion of the Universe -- the scales of interest are all deeply sub-horizon -- but the size of regions over which perturbations are evaluated far exceed $ct$, where $t$ is the time elapsed since the start of emission (it is assumed that $t$ greatly exceeds the duration of emission). We find that to lowest order, when only wave fields $\propto 1/r$ are included, there is exact compensation between the energy loss of the oscillators and the energy emitted into space, which means $P(0)=0$ for the power spectrum of density fluctuations on the largest scales. This is consistent with the Zel'dovich bound. To the next order when near fields $\propto r^{-2}$ are included, however, $P(0)$ settles at late times to a positive value that depends only on time, as $t^{-2}$ (the same applies to an energy non-conserving term). Even though this effect looks like superluminal energy transport, there is no violation of causality because the two-point function vanishes completely for $r>t$ if the emission of each oscillator is truncated beyond some duration. The result calls to question any need of enlisting cosmic inflation to seed large scale density perturbations. When applied to fast radio bursts -- uniformly distributed transients (to lowest order) that repeat at other locations -- the result supports Hoyle's hypothesis of constant energy injection.Comment: 17 pages, 38 equations, 2 appendices, final edited proof version, JCAP in pres

Exclusion of standard ℏω\hbar\omega gravitons by LIGO observation

Dyson (2013) argued that the extraordinarily large number of gravitons in a gravitational wave makes them impossible to be resolved as individual particles. While true, it is shown in this paper that a LIGO interferometric detector also undergoes frequent and {\it discrete} quantum interactions with an incident gravitational wave, in such a way as to allow the exchange of energy and momentum between the wave and the detector. This opens the door to another way of finding gravitons. The most basic form of an interaction is the first order Fermi acceleration (deceleration) of a laser photon as it is reflected by a test mass mirror oscillating in the gravitational wave, resulting in a frequency blueshift (redshift) of the photon depending on whether the mirror is advancing towards (receding from) the photon before the reflection. If e.g. a blueshift occurred, wave energy is absorbed and the oscillation will be damped. It is suggested that such energy exchanging interactions are responsible for the observed radiation reaction noise of LIGO (although the more common way of calculating the same amplitude for this noise is based on momentum considerations). Most importantly, in each interaction the detector absorbs or emits wave energy in amounts far smaller than the standard graviton energy $\hbar\omega$ where $\omega$ is the angular frequency of the gravitational wave. This sets a very tight upper limit on the quantization of the wave energy, viz. it must be at least $\approx 10^{11}$ times below $\hbar\omega$, independently of the value of $\omega$ itself.Comment: This final version was published on 17 August, 2018 by CQG as a Letter to the Edito

A microscopic derivation of Special Relativity: simple harmonic oscillations of a moving space-time lattice

The starting point of the theory of Special Relativity$^1$ is the Lorentz transformation, which in essence describes the lack of absolute measurements of space and time. These effects came about when one applies the Second Relativity Postulate to inertial observers. Here I demonstrate that there is a very elegant way of explaining how exactly nature enforces Special Relativity, which compels us to conclude that Einstein's great theory has already revealed quantization of space and time. The model proposes that microscopically the structure of space-time is analogous to a crystal which consists of lattice points or `tickmarks' (for measurements) connected by identical `elastic springs'. When at rest the `springs' are at their natural states. When set in motion and used to measure objects at rest, however, the lattice effectively vibrates in a manner described by Einstein's theory of the heat capacity of solids, with consequent widening of the `tickmarks' because the root-mean-square separation now increases. I associate a vibration temperature $T$ with the speed of motion $v$ via the fundamental postulate of this theory, viz. the relation $\frac{v^2}{c^2} = e^{-\frac{\epsilon}{kT}}$ where $\epsilon$ is a quantum of energy of the lattice harmonic oscillator. A moving observer who measures distances with such a vibrating lattice obtains results which are precisely those given by the Lorentz transformation. Apart from its obvious beauty, this approach provides many new prospects in understanding space and time. For example, a consequence of the model is that space-time, like mass, can in principle be converted to energy.Comment: 7 pages, 1 figur

Hubble redshift and the Heisenberg frequency uncertainty: on a coherence (or pulse) time signature in extragalactic light

In any Big Bang cosmology, the frequency $\omega$ of light detected from a distant source is continuously and linearly changing (usually redshifting) with elapsed observer's time $\delta t$, because of the expanding Universe. For small $\delta t$, however, the resulting $\delta\omega$ shift lies beneath the Heisenberg frequency uncertainty. And since there {\it is} a way of telling whether such short term shifts really exist, if the answer is affirmative we will have a means of monitoring radiation to an accuracy level that surpasses fundamental limitations. More elaborately, had $\omega$ been `frozen' for a minimum threshold interval before any redshift could take place, i.e. the light propagated as a smooth but {\it periodic} sequence of wave packets or pulses, and $\omega$ decreased only from one pulse to the next, one would then be denied the above forbiddingly precise information about frequency behavior. Yet because this threshold period is {\it observable}, being $\Delta t \approx 1/\sqrt{\omega_0 H_0} \sim$ 5 -- 15 minute for the cosmic microwave background (CMB), we can indeed perform a check for consistency between the Hubble Law and the Uncertainty Principle. If, as most would assume to be the case, the former either takes effect without violating the latter or not take effect at all, the presence of this characteristic time signature (periodicity) $\Delta t$ would represent direct verification of the redshift phenomenon.Comment: 9 pages, 18 equations, 1 figure. Paper overhauled. A much simpler interpretation of the CMB Fourier transform in terms of the Uncertainty Principle is availabl

Topological phases in the non-Hermitian Su-Schrieffer-Heeger model

We address the conditions required for a $\mathbb{Z}$ topological classification in the most general form of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. Any chirally-symmetric SSH model will possess a "conjugated-pseudo-Hermiticity" which we show is responsible for a quantized "complex" Berry phase. Consequently, we provide the first example where the complex Berry phase of a band is used as a quantized invariant to predict the existence of gapless edge modes in a non-Hermitian model. The chirally-broken, $PT$-symmetric model is studied; we suggest an explanation for why the topological invariant is a global property of the Hamiltonian. A geometrical picture is provided by examining eigenvector evolution on the Bloch sphere. We justify our analysis numerically and discuss relevant applications.Comment: 8 pages (PRB, in press

Harmonically dancing space-time nodes: quantitatively deriving relativity, mass, and gravitation

The microscopic structure of space and time is investigated. It is proposed that space and time of an inertial observer $\Sigma$ are most conveniently described as a crystal array $\Lambda$, with nodes representing measurement `tickmarks' and connected by independent quantized harmonic oscillators which vibrate more severely the faster $\Sigma$ moves with respect to the object being measured (due to the Uncertainty Principle). The Lorentz transformation of Special Relativity is derived. Further, mass is understood as a localized region $\Delta \Lambda$ having higher vibration temperature than that of the ambient lattice. The effect of relativistic mass increase may then be calculated without appealing to energy-momentum conservation. The origin of gravitation is shown to be simply a transport of energy from the boundary of $\Delta \Lambda$ outwards by lattice phonon conduction, as the system tends towards equilibrium. Application to a single point mass leads readily to the Schwarzschild metric, while a new solution is available for two point masses - a situation where General Relativity is too complicated to work with. The important consequence is that inertial observers who move at relative speeds too close to $c$ are no longer linked by the Lorentz transformation, because the lattice of the `moving' observer has already disintegrated into a liquid state.Comment: 13 pages, 3 figure

Relativity as the quantum mechanics of space-time measurements

Can a simple microscopic model of space and time demonstrate Special Relativity as the macroscopic (aggregate) behavior of an ensemble ? The question will be investigated in three parts. First, it is shown that the Lorentz transformation formally stems from the First Relativity Postulate (FRP) {\it alone} if space-time quantization is a fundamental law of physics which must be included as part of the Postulate. An important corollary, however, is that when measuring devices which carry the basic units of lengths and time (e.g. a clock ticking every time quantum) are `moving' uniformly, they appear to be measuring with larger units. Secondly, such an apparent increase in the sizes of the quanta can be attributed to extra fluctuations associated with motion, which are precisely described in terms of a thermally agitated harmonic oscillator by using a temperature parameter. This provides a stringent constraint on the microscopic properties of flat space-time: it is an array of quantized oscillators. Thirdly, since the foregoing development would suggest that the space-time array of an accelerated frame cannot be in thermal equilibrium, (i.e. it will have a distribution of temperatures), the approach is applied to the case of acceleration by the field of {\it any} point object, which corresponds to a temperature `spike' in the array. It is shown that the outward transport of energy by phonon conduction implies an inverse-square law of force at low speeds, and the full Schwarzschild metric at high speeds. A prediction of the new theory is that when two inertial observers move too fast relative to each other, or when fields are too strong, anharmonic corrections will modify effects like time dilation, and will lead to asymmetries which implies that the FRP may not be sustainable in this extreme limit.Comment: 17 pages, 3 figure

Has inflation really solved the problems of flatness and absence of relics?

Among the three cosmological enigma solved by the theory of inflation, {\it viz.} (a) large scale flatness, (b) absence of monopoles and strings, and (c) structure formation, the first two are addressed from the viewpoint of the observed scales having originated from very small ones, on which the density fluctuations of the curvaton and relics are {\it inevitably} of order unity or larger. By analyzing strictly classically (and in two different gauges to ensure consistency) the density evolution of the smoothest possible pre-inflationary component -- thermal radiation -- it is found that the O(1) statistical fluctuations on the thermal wavelength scale present formidable obstacles to the linear theory of amplitude growth by the end of inflation. Since this wavelength scale exited the horizon at an early stage of inflation, it severely limits the number of e-folds of perturbative inflation. With more e-folds than $\approx 60$ there will be even larger fluctuations in the radiation density that ensures inflation keeps making `false starts'. The only `way out' is to invoke a super-homogeneous pre-inflationary fluid, at least on small scales, adding to the fine-tuning and preventing one from claiming that inflation simply `redshifts away' all the relic inhomogeneities; {\it i.e} the theory actually provided no explanation of (a) or (b), merely a tautology.Comment: MNRAS in press (9 journal pages, 1 appendix); minor changes at proofs stage. Final versio

The outermost gravitationally bound orbit around a mass clump in an expanding Universe: implication on rotation curves and dark matter halo sizes

Conventional treatment of cold dark matter halos employs the Navarro-Frenk-White (NFW) profile with a maximum radius set at $r=r_{200}$, where the enclosed matter has an overdensity of 200 times the critical density. The choice of $r=r_{200}$ is somewhat arbitrary. It is not the collapsed (virial) radius, but does give $r \sim$ 1 Mpc for rich clusters, which is a typical X-ray size. Weak lensing measurements, however, reveal halo radii well in excess of $r_{200}$. Is there a surface that places an absolute limit on the extension of a halo? To answer the question, we derived analytically the solution for circular orbits around a mass concentration in an expanding flat Universe, to show that an outermost orbit exists at $v/r = H$, where $v$ is the orbital speed and $H$ is the Hubble constant. The solution, parametrized as $r_2$, is independent of model assumptions on structure formation, and {\it is the radius at which the furthest particle can be regarded as part of the bound system}. We present observational evidence in support of dark matter halos reaching at least as far out as $r=r_2$. An interesting consequence that emerges concerns the behavior of rotation curves. Near $r=r_2$ velocities will be biased low. As a result, the mass of many galaxy groups may have been underestimated. At $r=r_2$ there is an abrupt cutoff in the curve, irrespective of the halo profile. An important cosmological test can therefore be performed if velocity disperion data are available out to 10 Mpc radii for nearby clusters (less at higher redshifts). For Virgo it appears that there is no such cutoff.Comment: Shortened to satisfy ApJL page limit. Re-submitted. First version contains serious error due to cancellation of lowest order Hubble effect in the Lab. 2nd order effect is genuine, and is presented her

Proving the conservation of surface brightness during the strong and weak lensing of light by an isothermal sphere

An analytical proof of the conservation of surface brightness during the strong and weak lensing of light by a singular isothermal sphere is provided. It is shown that the movement of asymptotic rays provide room for precisely the extra solid angle claimed by the magnification of the centrally passing rays. Previous claim of a violation of this conservation law, leading to a problem over the COBE all sky CMB flux, is hereby withdrawn.Comment: Research note (for pedagogical purpose