Spatial Flatness Research Articles

Research on a Computerized Form and Position Error Measurement Instrument

A model machine of multifunctional form and position measurement instrument controlled by a personal computer has been successfully developed. The instrument is designed in rotary table type with a high precision air bearing and the radial rotation error of the rotary table is 0.08 μm. A high precision vertical sliding carriage supported by an air bearing is used to the instrument, the straight motion error of the carriage is 0.3 μm/200 mm and the parallelism error of the motion of the carriage relative to the rotation axis of the rotary table is 0.4 μm/200 mm. The mathematical models have been established for assessing planar and spatial straightness, flatness, roundness, cylindricity, and coaxality errors. By radial deviation measurement, the instrument can accurately measure form and position errors of such workpieces as shafts, round plates and sleeves of medium or small dimensions with the tolerance grades mostly used in industry.

Dynamics of Multiple Scalar Fields Model of Dark Energy with Spatial Curvature

The dynamical system of multiple scalar fields in FRW universe with different spatial curvature have been analyzed in this paper. In the radiation-dominated phase, the constant curvature factor k does not work on the cosmic dynamical behaviors, including the scaling solution, energy density parameter and equation-of-state parameter. These aspects are affected by curvature factor k in the matter-dominated phase. In the special scalar field-dominated phase, the energy density parameter normalization restricts the Universe is spatial flat and the curvature factor k is not present in the dynamics. In this paper, the Universe is closed in the matter-dominated phase, and flat in the scalar field-dominated phase. The spatial flatness and the wϕ=−1 in the third phase are coincide with the current observations.

Cosmogenesis and the tipping pencil analogy

We present a toy model and scenario for cosmogenesis which provides an interesting rationale for the spatial flatness of the universe. The basic assumptions are that the cosmological scale factor obeys the standard Friedmann equation of general relativistic cosmology and that the equation is dominated by a cosmological constant term and a curvature term. The dynamics of the universe is then similar to that of a tipping pencil. The scale factor cannot remain at an unstable initial value of zero and must increase according to the uncertainty principle. If it is also assumed that the universe expands at the minimum asymptotic rate consistent with the uncertainty principle, the result is exact spatial flatness, rather than the near flatness obtained in standard inflation theory.

On topological modifications of Newton's law

Recent cosmological data for very large distances challenge the validity of the standard cosmological model. Motivated by the observed spatial flatness the accelerating expansion and the various anisotropies with preferred axes in the universe we examine the consequences of the simple hypothesis that the three-dimensional space has a global ℛ2 × \U0001d4ae1 topology. We take the radius of the compactification to be the observed cosmological scale beyond which the accelerated expansion starts. We derive the induced corrections to the Newton's gravitational potential and we find that for distances smaller than the \U0001d4ae1 radius the leading 1/r-term is corrected by convergent power series of multipole form in the polar angle making explicit the induced anisotropy by the compactified third dimension. On the other hand, for distances larger than the compactification scale the asymptotic behavior of the potential exhibits a logarithmic dependence with exponentially small corrections. The change of Newton's force from 1/r2 to 1/r behavior implies a weakening of the deceleration for the expanding universe. Such topologies can also be created locally by standard Newtonian axially symmetric mass distributions with periodicity along the symmetry axis. In such cases we can use our results to obtain measurable modifications of Newtonian orbits for small distances and flat rotation spectra, for large distances at the galactic level.

Gravitational waves and non-axisymmetric oscillation modes in mergers of compact object binaries

We study the excitation of nonaxisymmetric modes in the post-merger phase of binary compact object mergers and the associated gravitational wave emission. Our analysis is based on general-relativistic simulations, in the spatial conformal flatness approximation, using smoothed-particle-hydrodynamics for the evolution of matter, and we use a set of equal and unequal mass models, described by two nonzero-temperature hadronic equations of state and by one strange star equation of state. Through Fourier transforms of the evolution of matter variables, we can identify a number of oscillation modes, as well as several nonlinear components (combination frequencies). We focus on the dominant m=2 mode, which forms a triplet with two nonlinear components that are the result of coupling to the quasiradial mode. A corresponding triplet of frequencies is identified in the gravitational wave spectrum, when the individual masses of the compact objects are in the most likely range of 1.2 to 1.35 $M_\odot$. We can thus associate, through direct analysis of the dynamics of the fluid, a specific frequency peak in the gravitational wave spectrum with the nonlinear component resulting from the difference between the m=2 mode and the quasiradial mode. Once such observations become available, both the m=2 and quasiradial mode frequencies could be extracted, allowing for the application of gravitational-wave asteroseismology to the post-merger remnant and leading to tight constraints on the equation of state of high-density matter.

A conventional approach to the dark-energy concept

Motivated by results implying that the constituents of dark matter (DM) might be collisional, we consider a cosmological (toy-) model, in which the DM itself possesses some sort of thermodynamic properties. In this case, not only can the matter content of the Universe be treated as a classical gravitating fluid of positive pressure, but, together with all its other physical characteristics, the energy of this fluid's internal motions should be taken into account as a source of the universal gravitational field. This form of energy can compensate for the extra (dark) energy, needed to compromise spatial flatness, while the post-recombination Universe remains ever-decelerating. At the same time (i.e., in the context of the collisional-DM approach), the theoretical curve representing the distance modulus as a function of the cosmological redshift, {\mu}(z), fits the Hubble diagram of a multi-used sample of supernova Ia events quite accurately. A cosmological model filled with collisional DM could accommodate the majority of the currently-available observational data (including, also, those from baryon acoustic oscillations), without the need for either any dark energy (DE) or the cosmological constant. However, as we demonstrate, this is not the case for someone who, although living in a Universe filled with self-interacting DM, insists on adopting the traditional, collisionless-DM approach. From the point of view of this observer, the cosmologically-distant light-emitting sources seem to lie farther (i.e., they appear to be dimmer) than expected, while the Universe appears to be either accelerating or decelerating, depending on the value of the cosmological redshift. This picture, which, nowadays, represents the common perception in observational cosmology, acquires a more conventional interpretation within the context of the collisional-DM approach.

Neutrino constraints from future nearly all-sky spectroscopic galaxy surveys

We examine whether future, nearly all-sky galaxy redshift surveys, in combination with CMB priors, will be able to detect the signature of the cosmic neutrino background and determine the absolute neutrino mass scale. We also consider what constraints can be imposed on the effective number of neutrino species. In particular we consider two spectroscopic strategies in thenear-IR, the so-called ``slitless'' and ``multi-slit'' approaches, whose examples are givenby future space-based galaxy surveys, as EUCLID for the slitless case,or SPACE, JEDI, and possibly WFIRST in the future, for the multi-slit case.We find that, in combination with Planck, these galaxy probes will be able to detect at better than 3-sigma level and measure the mass of cosmic neutrinos: a) in a cosmology-independent way, if the sum of neutrino masses is above 0.1 eV; b) assuming spatial flatness and that dark energy is a cosmological constant, otherwise. We find that the sensitivity of such surveys is well suited to span the entire range of neutrino masses allowed by neutrino oscillation experiments,and to yield a clear detection of non-zero neutrino mass.The detection of the cosmic relic neutrino background withcosmological experiments will be a spectacular confirmation of our model for the early Universe and a window into one of the oldest relic components of our Universe.

A conventional form of dark energy

Motivated by recent results, indicating that the dark matter (DM) constituents can be collisional, we assume that the DM itself possesses also some sort of thermodynamical properties. In this case, the Universe matter-content can be treated as a gravitating fluid of positive pressure, and, therefore, together with all the other physical characteristics, the energy of this fluid's internal motions should be taken into account as a source of the universal gravitational field. In principle, this form of energy can compensate, also, the extra (dark) energy, needed to compromise spatial flatness, while, the post-recombination Universe remains ever-decelerating. What is more interesting, is that, at the same time (i.e., in the context of the collisional-DM approach), the theoretical curve, representing the distance modulus as a function of the cosmological redshift, fits the Hubble diagram of an extended sample of SN Ia events quite accurately. However, as we demonstrate, this is not the case for someone who, although living in a Universe filled with collisional DM, insists in adopting the traditional, collisionless-DM approach. From the point of view of such an observer, the distant light-emitting sources seem to lie farther (i.e., they appear to be dimmer) than expected, while, the Universe appears to be either accelerating or decelerating, depending on the value of the cosmological redshift.

On the accelerated expansion of the cosmos

We present a short (and necessarily incomplete) review of the evidence for the accelerated expansion of the Universe. The most direct probe of acceleration relies on the detailed study of supernovae (SN) of type Ia. Assuming that these are standardizable candles and that they fairly sample a homogeneous and isotropic Universe, the evidence for acceleration can be tested in a model-independent and calibration-independent way. Various light-curve fitting procedures have been proposed and tested. While several fitters give consistent results for the so-called Constitution set, they lead to inconsistent results for the recently released SDSS SN. Adopting the SALT fitter and relying on the Union set, cosmic acceleration is detected by a purely kinematic test at 7 σ when spatial flatness is assumed and at 4 σ without any assumption on the spatial geometry. A weak point of the described method is the local set of SN (at z < 0.2 ), as these SN are essential to anchor the Hubble diagram. These SN are drawn from a volume much smaller than the Hubble volume and could be affected by local structure. Without the assumption of homogeneity, there is no evidence for acceleration, as the effects of acceleration are degenerate with the effects of inhomogeneities. Unless we sit in the centre of the Universe, such inhomogeneities can be constrained by SN observations by means of tests of the isotropy of the Hubble flow.

Nonconformally flat initial data for binary compact objects

A new method is described for constructing initial data for a binary neutron-star (BNS) system in quasi-equilibrium circular orbit. Two formulations for non-conformally flat data, waveless (WL) and near-zone helically symmetric (NHS), are introduced; in each formulation, the Einstein-Euler system, written in 3+1 form on an asymptotically flat spacelike hypersurface, is exactly solved for all metric components, including the spatially non-conformally flat potentials, and for irrotational flow. A numerical method applicable to both formulations is explained with an emphasis on the imposition of a spatial gauge condition. Results are shown for solution sequences of irrotational BNS with matter approximated by parametrized equations of state that use a few segments of polytropic equations of state. The binding energy and total angular momentum of solution sequences computed within the conformally flat -- Isenberg-Wilson-Mathews (IWM) -- formulation are closer to those of the third post-Newtonian (3PN) two point particles up to the closest orbits, for the more compact stars, whereas sequences resulting from the WL/NHS formulations deviate from the 3PN curve even more for the sequences with larger compactness. We think it likely that this correction reflects an overestimation in the IWM formulation as well as in the 3PN formula, by $\sim 1$ cycle in the gravitational wave phase during the last several orbits. The work suggests that imposing spatial conformal flatness results in an underestimate of the quadrupole deformation of the components of binary neutron-star systems in the last few orbits prior to merger.

Anti–de Sitter 5D black hole solutions with a self-interacting bulk scalar field: A potential reconstruction approach

We construct asymptotically AdS black hole solutions, with a self-interacting bulk scalar field, in the context of 5D general relativity. As the observable universe is characterized by spatial flatness, we focus on solutions where the horizon of the black hole, and subsequently all 3D hypersurfaces for fixed radial coordinate, have zero spatial curvature. We examine two cases for the black hole scalar hair: (a) an exponential decaying scalar field profile and (b) an inverse power scalar field profile. The scalar black hole solutions we present in this paper are characterized by four functions $f(r)$, $a(r)$, $\ensuremath{\phi}(r)$, and $V(\ensuremath{\phi}(r))$. Only the functions $\ensuremath{\phi}(r)$ and $a(r)$ are determined analytically, while the functions $f(r)$ and $V(\ensuremath{\phi}(r))$ are expressed semianalytically by integral formulas in terms of $a(r)$. We present our numerical results and study in detail the characteristic properties of our solutions. We also note that the potential we obtain has a nonconvex form in agreement with the corresponding ``no hair theorem'' for AdS spacetimes.

Dark energy or apparent acceleration due to a relativistic cosmological model more complex than the Friedmann-Lemaitre-Robertson-Walker model?

We use the Szekeres inhomogeneous relativistic models in order to fit supernova combined data sets. We show that with a choice of the spatial curvature function that is guided by current observations, the models fit the supernova data almost as well as the Lambda-CDM model without requiring a dark energy component. The Szekeres models were originally derived as an exact solution to Einstein's equations with a general metric that has no symmetries, and they are regarded as good candidates to model the true lumpy universe that we observe. The null geodesics in these models are not radial. The best fit model found is also consistent with the requirement of spatial flatness at cosmic microwave background scales. The first results presented here seem to encourage further investigations of apparent acceleration using various inhomogeneous models, and other constraints from cosmic microwave background and large structure need to be explored next.

Cosmic dynamics in the era of Extremely Large Telescopes

The redshifts of all cosmologically distant sources are expected to experience a small, systematic drift as a function of time due to the evolution of the Universe's expansion rate. A measurement of this effect would represent a direct and entirely model-independent determination of the expansion history of the Universe over a redshift range that is inaccessible to other methods. Here we investigate the impact of the next generation of Extremely Large Telescopes on the feasibility of detecting and characterising the cosmological redshift drift. We consider the Lyman alpha forest in the redshift range 2 < z < 5 and other absorption lines in the spectra of high redshift QSOs as the most suitable targets for a redshift drift experiment. Assuming photon-noise limited observations and using extensive Monte Carlo simulations we determine the accuracy to which the redshift drift can be measured from the Ly alpha forest as a function of signal-to-noise and redshift. Based on this relation and using the brightness and redshift distributions of known QSOs we find that a 42-m telescope is capable of unambiguously detecting the redshift drift over a period of ~20 yr using 4000 h of observing time. Such an experiment would provide independent evidence for the existence of dark energy without assuming spatial flatness, using any other cosmological constraints or making any other astrophysical assumption.

Relativistic black hole-neutron star binaries in quasiequilibrium: Effects of the black hole excision boundary condition

We construct new models of black hole-neutron star binaries in quasiequilibrium circular orbits by solving Einstein's constraint equations in the conformal thin-sandwich decomposition together with the relativistic equations of hydrostationary equilibrium. We adopt maximal slicing, assume spatial conformal flatness, and impose equilibrium boundary conditions on an excision surface (i.e., the apparent horizon) to model the black hole. In our previous treatment we adopted a ``leading-order'' approximation for a parameter related to the black hole spin in these boundary conditions to construct approximately nonspinning black holes. Here we improve on the models by computing the black hole's quasilocal spin angular momentum and setting it to zero. As before, we adopt a polytropic equation of state with adiabatic index $\ensuremath{\Gamma}=2$ and assume the neutron star to be irrotational. In addition to recomputing several sequences for comparison with our earlier results, we study a wider range of neutron star masses and binary mass ratios. To locate the innermost stable circular orbit we search for turning points along both the binding energy and total angular momentum curves for these sequences. Unlike for our previous approximate boundary condition, these two minima now coincide. We also identify the formation of cusps on the neutron star surface, indicating the onset of tidal disruption. Comparing these two critical binary separations for different mass ratios and neutron star compactions we distinguish those regions that will lead to a tidal disruption of the neutron star from those that will result in the plunge into the black hole of a neutron star more or less intact, albeit distorted by tidal forces.

Consequences of a Cosmological Phase Transition at the TeV Scale

A finite vacuum energy density implies the existence of a UV scale for gravitational modes. This gives a phenomenological scale to the dynamical equations governing the cosmological expansion that must satisfy constraints consistent with quantum measurability and spatial flatness. Examination of these constraints for the observed dark energy density establishes a time interval from the transition to the present, suggesting major modifications from the thermal equations of state far from Planck density scales. The assumption that a phase transition initiates the radiation dominated epoch is shown under several scenarios to be able to produce fluctuations to the CMB of the order observed. Quantum measurability constraints (eg. uncertainly relations) define cosmological scales bounded by luminal expansion rates. It is shown that the dark energy can consistently be interpreted as being due to the vacuum energy of collective gravitational modes which manifest as the zero-point motions of coherent Planck scale mass units prior to the UV scale onset of gravitational quantum de-coherence for the cosmology. A cosmological model with multiple scales, one of which replaces an apparent cosmological “constant”, is shown to reproduce standard cosmology during intermediate times, while making the exploration of the early and late time cosmology more accessible.

Development of a Computerized Multifunctional Form and Position Measurement Instrument

A model machine of multifunctional form and position measurement instrument controlled by a personal computer has been successfully developed. The instrument is designed in rotary table type with a high precision air bearing and the radial rotation error of the rotary table is 0.08 μm. Since a high precision vertical sliding carriage supported by an air bearing is used for the instrument, the straightaway motion error of the carriage is 0.3 μm/200 mm and the parallelism error of the motion of the carriage relative to the rotation axis of the rotary table is 0.4 μm/200 mm. The mathematical models have been established for assessing planar and spatial straightness, flatness, roundness, cylindricity, and coaxality errors. By radial deviation measurement, the instrument can accurately measure form and position errors of such workpieces as shafts, round plates and sleeves of medium or small dimensions with the tolerance grades mostly used in industry.

Power spectrum of the SDSS luminous red galaxies: constraints on cosmological parameters

In this paper we determine the constraints on cosmological parameters using the CMB data from the Wmap experiment together with the recent power spectrum measurement of the SDSS Luminous Red Galaxies (LRGs). Specifically, we focus on spatially flat, low matter density models with adiabatic Gaussian initial conditions. The spatial flatness is achieved with an additional quintessence component whose equation of state parameter $w_\mathrm{eff}$ is taken to be independent of redshift. Throughout most of the paper we do not allow any massive neutrino contribution and also the influence of the gravitational waves on the CMB is taken to be negligible. The analysis is carried out separately for two cases: (i) using the acoustic scale measurements as presented in Hutsi (2006, A&A, 449, 891), (ii) using the full SDSS LRG power spectrum and its covariance matrix. We are able to obtain a very tight constraint on the Hubble constant: $H_0 = 70.8 ^{+2.1}_{-2.0}$ km s -1 Mpc -1 , which helps in breaking several degeneracies between the parameters and allows us to determine the low redshift expansion law with much higher accuracy than available from the Wmap + HST data alone. The positive deceleration parameter q 0 is found to be ruled out at $5.5\sigma$ confidence level. Finally, we extend our analysis by investigating the effects of relaxing the assumption of spatial flatness and also allow for a contribution from massive neutrinos.

Curved space or curved vacuum?

While the simple picture of a spatially flat, matter plus cosmological constant universe fits current observation of the accelerated expansion, strong consideration has also been given to models with dynamical vacuum energy. We examine the tradeoff of “curving” the vacuum but retaining spatial flatness, vs. curving space but retaining the cosmological constant. These different breakdowns in the simple picture could readily be distinguished by combined high accuracy supernovae and cosmic microwave background distance measurements. If we allow the uneasy situation of both breakdowns, the curvature can still be measured to 1%, but at the price of degrading estimation of the equation of state time variation by 60% or more, unless additional information (such as weak lensing data or a tight matter density prior) is included.

Uncorrelated measurements of the cosmic expansion history and dark energy from supernovae

We present a method for measuring the cosmic expansion history $H(z)$ in uncorrelated redshift bins, and apply it to current and simulated type Ia supernova data assuming spatial flatness. If the matter density parameter ${\ensuremath{\Omega}}_{m}$ can be accurately measured from other data, then the dark-energy density history $X(z)={\ensuremath{\rho}}_{X}(z)/{\ensuremath{\rho}}_{X}(0)$ can trivially be derived from this expansion history $H(z)$. In contrast to customary ``black box'' parameter fitting, our method is transparent and easy to interpret: the measurement of $H(z{)}^{\ensuremath{-}1}$ in a redshift bin is simply a linear combination of the measured comoving distances for supernovae in that bin, making it obvious how systematic errors propagate from input to output. We find the Riess et al. (2004) gold sample to be consistent with the vanilla concordance model where the dark energy is a cosmological constant. We compare two mission concepts for the NASA/DOE Joint Dark-Energy Mission (JDEM), the Joint Efficient Dark-energy Investigation (JEDI), and the Supernova Accelaration Probe (SNAP), using simulated data including the effect of weak lensing (based on numerical simulations) and a systematic bias from K corrections. Estimating $H(z)$ in seven uncorrelated redshift bins, we find that both provide dramatic improvements over current data: JEDI can measure $H(z)$ to about 10% accuracy and SNAP to 30%--40% accuracy.

Automatic hyperparameter estimation in probabilistic image restoration algorithm based on Gibbs microcanonical distribution

For an original image given as an 8-gray-level image, the framework for image restoration is composed of Bayes statistics for the a priori probability distribution based on the microcanonical distribution. In this process, the a priori probability distribution serving as the microcanonical distribution is formulated on the basis of the variables characterizing the spatial flatness and smoothness of the image in terms of the number of nearest pixel pairs with different gray levels and the number of nearest pixel pairs with gray levels differing by 1. In the present paper, we propose a new method of estimation, from a degraded image, of the number of nearest pixel pairs with different gray levels and the number of nearest pixel pairs with gray levels differing by 1 in the original image. Then, based on that formulation, an iterative computation algorithm for the restoration of a 256-gray-level image is presented from the viewpoint of statistical mechanics. Through some numerical experiments, we investigate how the proposed method can improve the quality of the restored image. © 2004 Wiley Periodicals, Inc. Syst Comp Jpn, 36(1): 68–78, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10365