Deceleration parameter

The redshift-magnitude or z-m diagram (Fig. A2.2) has become a standard tool in cosmology because it can be used to determine past variations in the expansion rate, through what is usually called the deceleration parameter. Any acceleration or slowdown of the expansion can thus be brought to light. 

Here, we shall discuss the implications of cosmological expansion for the searches of a quantum-gravity-induced refractive index and a stochastic effect. We will consider Friedman-Robertson-Walker (FRW) metrics as an appropriate candidate for standard homogeneous and isotropic cosmology. Let R be the FRW scale factor, and a subscript 0 will denote the value at the present era. Ho is the present Hubble expansion parameter, and the deceleration parameter qo is defined in terms of the curvature k of the FRW metric by k ( 2[Pg.588]

In the past decade several projects contributed to the luminosity distance measurements and by now (i.e., as of 2009) the list includes over 200 events. Specifically with the help of the Hubble telescope 13 new Sn la were found with spectroscopically confirmed redshifts exceeding z = 1 and at present the full sample contains already 23 z > 1 objects (Riess et al. 2007). Such objects most strongly influence the value of the deceleration parameter. A combined analysis of all Sn la data yields a deceleration parameter value of —0.7 0.1 (Kowalski et al. 2008). Its negative value signals an accelerating expansion rate at distance scales comparable to the size of the Universe. 

In the Friedmann cosmologies of the GTR, a finite space implies an end to proper time in the future, but this is not required in some nonstandard models. Assuming the Robertson-Walker form of the metric for a homogeneous and isotropic space-time, it is convenient to discuss the future evolution of any such expanding universe in terms of a dimensionless deceleration parameter, defined as ... 

In their works,51"54 the self-similar fractal dimension dF>ss of the two-dimensional distribution of the pits was determined by the analysis of the digitized SEM images using the perimeter-area method. The value of dF>ss increased with increasing solution temperature,51 and it was inversely proportional to the pit shape parameter and the pit growth rate parameter.53 Keeping in mind that dr>ss is inversely proportional to the increment of the pit area density, these results can be accounted for in terms of the fact that the increment of the pit area density is more decelerated with rising solution temperature. 

As an example, let us analyze mold filling with a model polyurethane formulation. Let the kinetics of curing be described by an equation with the self-deceleration term (as was discussed above). The following values of the parameters were used U = 49.1 kJ/mol ko = 3.8xl06 = 1.1 ATmax = 25.8°C where ATmax is the maximum expected increase in temperature for adiabatic curing. 

Based mainly on the analytical results for single particle motion in impinging streams, Tamir derived a number of expressions for the two parameters for various flow regimes in the two cases with and without chemical reaction, in which the parameters such as the droplet size, the motion times of a particle in the accelerating and decelerating stages, particle to gas velocity ratio at the outlet of the accelerating tube, etc. were involved (see Eqs. 11.2 to 11.25 in Ref. [5]). 

When particles are accelerated in a gas, their motion is governed by the balance between inertial, viscous, and external forces. An important characteristic scale is the time for an accelerated particle to achieve steady motion. To find this parameter, the deceleration of a particle by friction in a stationary gas is considered. In the absence of external forces, the velocity of a particle (q) traveling in the x direction is calculated by ... 

The observed deceleration (Table 20) is then essentially attributed to steric interaction in the transition state between the 2- or 4-alkyl group and the entering methylating agent. A good correlation between log k/kMe and the Taft parameter Es has been found for both series (equation 4). 

Similar rate decelerations in liquid-crystalline solvents have been observed for the thermal cis trans isomerization of a bulky tetrasubstituted ethene in cholesteric phases [728], On the other hand, the activation parameters for the thermal cis trans isomerization of less-dipolar substituted azobenzenes show no dependence on the solvent order. This indicates that the cis isomers and their corresponding activated complexes present a similar steric appearance to the solvent environment [729]. This result is more consistent with an isomerization mechanism which proceeds by inversion rather than by rotation cf. Eq. (5-40) in Section 5.3.2 and [527-529, 561]. The latter reaction represents a nice example of the use of liquid-crystalline solvents as mechanistic probes [729]. 

The effect of j5-cyclopropyl groups on hydration according to equation 40 was demonstrated to be decelerating relative to hydrogen or methyl, as shown by the data in Table 21. This effect was interpreted in terms of the transition state 174, in which the rate effect of P-c-Ft was correlated with a term proportional to the cr parameter of cyclopropyl (— 0.04) and another term proportional to the estimated stabilizing effect of cyclopropyl on a double bond (4.4 kcal mol ) . 

The parameter of the EPR 6, 0 < 6 < 1, characterizes the portion of the duct s cross-section occupied by the EPR. The calculations show that its increase does not decelerate the frequently oscillating medium which also differs from the low-frequency case. The effects of the easily penetrable roughness on a pulsating flow can be classified, in summary, depending on either the EPR is used for controlling the pulsating flow or, conversely, pressure pulsations are used to influence the flow in a duct with the EPR ... 

The advancement of a flow through the duct is accompanied by its deceleration near the walls and by the EPR drag force, thus displacing the liquid to the center of the duct. This leads to the surprising maxima on the longitudinal velocity profiles, Fig. 3.12. The flow in the middle constantly accelerates, but its value tends to a certain limit reached in the main steady-state region. For this laminar case, v = const, the dimensionless axial velocity is known to reach the value 1.5 irrespective to Re [380], if the EPR is absent. In the case of interest, it depends upon all the parameters A, 6, and Re, can be... 

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