A test of the nature of cosmic acceleration using galaxy redshift distortions.

Observations of distant supernovae indicate that the Universe is now in a phase of accelerated expansion the physical cause of which is a mystery. Formally, this requires the inclusion of a term acting as a negative pressure in the equations of cosmic expansion, accounting for about 75 per cent of the total energy density in the Universe. The simplest option for this 'dark energy' corresponds to a 'cosmological constant', perhaps related to the quantum vacuum energy. Physically viable alternatives invoke either the presence of a scalar field with an evolving equation of state, or extensions of general relativity involving higher-order curvature terms or extra dimensions. Although they produce similar expansion rates, different models predict measurable differences in the growth rate of large-scale structure with cosmic time. A fingerprint of this growth is provided by coherent galaxy motions, which introduce a radial anisotropy in the clustering pattern reconstructed by galaxy redshift surveys. Here we report a measurement of this effect at a redshift of 0.8. Using a new survey of more than 10,000 faint galaxies, we measure the anisotropy parameter beta = 0.70 +/- 0.26, which corresponds to a growth rate of structure at that time of f = 0.91 +/- 0.36. This is consistent with the standard cosmological-constant model with low matter density and flat geometry, although the error bars are still too large to distinguish among alternative origins for the accelerated expansion. The correct origin could be determined with a further factor-of-ten increase in the sampled volume at similar redshift.

A large population of galaxies 9 to 12 billion years back in the history of the Universe.

To understand the evolution of galaxies, we need to know as accurately as possible how many galaxies were present in the Universe at different epochs. Galaxies in the young Universe have hitherto mainly been identified using their expected optical colours, but this leaves open the possibility that a significant population remains undetected because their colours are the result of a complex mix of stars, gas, dust or active galactic nuclei. Here we report the results of a flux-limited I-band survey of galaxies at look-back times of 9 to 12 billion years. We find 970 galaxies with spectroscopic redshifts between 1.4 and 5. This population is 1.6 to 6.2 times larger than previous estimates, with the difference increasing towards brighter magnitudes. Strong ultraviolet continua (in the rest frame of the galaxies) indicate vigorous star formation rates of more than 10-100 solar masses per year. As a consequence, the cosmic star formation rate representing the volume-averaged production of stars is higher than previously measured at redshifts of 3 to 4.

Mechanics of competition walking.

1. The work done at each step to lift and accelerate the centre of mass of the body has been measured in competition walkers during locomotion from 2 to 20 km/hr. 2. Three distinct phases characterize the mechanics of walking. From 2 to 6 km/hr the vertical displacement during each step, Sv, increases to a maximum (3.5 vs. 6 cm in normal walking) due to an increase in the amplitude of the rotation over the supporting leg. 3. The transfer, R, between potential energy of vertical displacement and kinetic energy of forward motion during this rotation, reaches a maximum at 4-5 km/hr (R = 65%). From 6 to 10 km/hr R decreases more steeply than in normal walking, indicating a smaller utilization of the pendulum-like mechanism characteristic of walking. 4. Above 10 km/hr potential and kinetic energies vary during each step because both are simultaneously taken up and released by the muscles with almost no transfer between them (R = 2-10%). Above 13-14 km/hr an aerial phase (25-60 msec) takes place during the step. 5. Speeds considerably greater than in normal walking are attained thanks to a greater efficiency of doing positive work. This is made possible by a mechanism of locomotion allowing an important storage and recovery of mechanical energy by the muscles.

The determinants of the step frequency in walking in humans.

The mechanical power spent during walking in lifting and accelerating the centre of mass, Wext, has been measured at three given speeds maintained at different step frequencies: at any given speed, Wext is smaller the greater the step frequency used. The mechanical power spent in accelerating the limbs relative to the centre of mass during walking at a given speed, but with different step frequencies, Wint, was calculated from previous data obtained during free walking (Cavagna & Kaneko, 1977). At a given walking speed, Wint increases with the step frequency. The total power, Wtot = Wext + Wint, reaches a minimum at a step frequency which is 20-30% less than the step frequency freely chosen at the same period. The step frequency at which Wtot is minimum increases with speed in a similar way to the natural step frequency during free walking.

The mechanics of walking in children.

The work done at each step, during level walking at a constant average speed, to lift the centre of mass of the body, to accelerate it forward, and to increase the sum of both gravitational potential and kinetic energies, has been measured at various speeds on children of 2-12 years of age, with the same technique used previously for adults (Cavagna, 1975; Cavagna, Thys & Zamboni, 1976). The pendulum-like transfer between potential and kinetic energies (Cavagna et al. 1976) reaches a maximum at the speed at which the weight-specific work to move the centre of mass a given distance is at a minimum ('optimal' speed). This speed is about 2 X 8 km/hr at 2 years of age and increases progressively with age up to 5 km/hr at 12 years of age and in adults. The speed freely chosen during steady walking at the different ages is similar to this 'optimal' speed. At the 'optimal' speed, the time of single contact (time of swing) is in good agreement with that predicted, for the same stature, by a ballistic walking model assuming a minimum of muscular work (Mochon & McMahon, 1980). Above the 'optimal' speed, the recovery of mechanical energy through the potential-kinetic energy transfer decreases. This decrease is greater the younger the subject. A reduction of this recovery implies a greater amount of work to be supplied by muscles: at 4 X 5 km/hr the weight-specific muscular power necessary to move the centre of mass is 2 X 3 times greater in a 2-year-old child than in an adult.

The determinants of the step frequency in running, trotting and hopping in man and other vertebrates.

1. During each step of running, trotting or hopping part of the gravitational and kinetic energy of the body is absorbed and successively restored by the muscles as in an elastic rebound. In this study we analysed the vertical motion of the centre of gravity of the body during this rebound and defined the relationship between the apparent natural frequency of the bouncing system and the step frequency at the different speeds. 2. The step period and the vertical oscillation of the centre of gravity during the step were divided into two parts: a part taking place when the vertical force exerted on the ground is greater than body weight (lower part of the oscillation) and a part taking place when this force is smaller than body weight (upper part of the oscillation). This analysis was made on running humans and birds; trotting dogs, monkeys and rams; and hopping kangaroos and springhares. 3. During trotting and low-speed running the rebound is symmetric, i.e. the duration and the amplitude of the lower part of the vertical oscillation of the centre of gravity are about equal to those of the upper part. In this case, the step frequency equals the frequency of the bouncing system. 4. At high speeds of running and in hopping the rebound is asymmetric, i.e. the duration and the amplitude of the upper part of the oscillation are greater than those of the lower part, and the step frequency is lower than the frequency of the system. 5. The asymmetry is due to a relative increase in the vertical push. At a given speed, the asymmetric bounce requires a greater power to maintain the motion of the centre of gravity of the body, Wext, than the symmetric bounce. A reduction of the push would decrease Wext but the resulting greater step frequency would increase the power required to accelerate the limbs relative to the centre of gravity, Wint. It is concluded that the asymmetric rebound is adopted in order to minimize the total power, Wext + Wint.

The two power limits conditioning step frequency in human running.

1. At high running speeds, the step frequency becomes lower than the apparent natural frequency of the body's bouncing system. This is due to a relative increase of the vertical component of the muscular push and requires a greater power to maintain the motion of the centre of gravity, Wext. However, the reduction of the step frequency leads to a decrease of the power to accelerate the limbs relatively to the centre of gravity, Wint, and, possibly, of the total power Wtot = Wext + Wint. 2. In this study we measured Wext using a force platform, Wint by motion picture analysis, and calculated Wtot during human running at six given speeds (from 5 to 21 km h-1) maintained with different step frequencies dictated by a metronome. The power was calculated by dividing the positive work done at each step by the duration of the step (step-average power) and by the duration of the positive work phase (push-average power). 3. Also in running, as in walking, a change of the step frequency at a given speed has opposite effects on Wext, which decreases with increasing step frequency, and Wint, which increases with frequency; in addition, a step frequency exists at which Wtot reaches a minimum. However, the frequency for a minimum of Wtot decreases with speed in running, whereas it increases with speed in walking. This is true for both the step-average and the push-average powers. 4. The frequency minimizing the step-average power equals the freely chosen step frequency at about 13 km h-1: it is higher at lower speeds and lower at higher speeds. The frequency minimizing the push-average power approaches the freely chosen step frequency at high speeds (around 22 km h-1 for our subjects). 5. It is concluded that the increase of the vertical push does reduce the step-average power, but that a limit is set by the increase of the push-average power. Between 13 and 22 km h-1 the freely chosen step frequency is intermediate between a frequency minimizing the step-average power, eventually limited by the maximum oxygen intake (aerobic power), and a frequency minimizing the push-average power, set free by the muscle immediately during contraction (anaerobic power). The first need prevails at the lower speed, the second at the higher speed.