Today, f (T ) theory has been one of the popular modified gravity theories to explain the accelerated expansion of the universe without invoking dark energy. In this work, we consider the so-called Hojman symmetry in f (T ) theory. Unlike Noether conservation theorem, the symmetry vectors and the corresponding conserved quantities in Hojman conservation theorem can be obtained by using directly the equations of motion, rather than Lagrangian or Hamiltonian. We find that Hojman symmetry can exist in f (T ) theory, and the corresponding exact cosmological solutions are obtained. We find that the functional form of f (T ) is restricted to be the power-law or hypergeometric type, while the universe experiences a power-law or hyperbolic expansion. These results are different from the ones obtained by using Noether symmetry in f (T ) theory. Therefore, it is reasonable to find exact cosmological solutions via Hojman symmetry.

In this perspective, we outline that a space borne gravitational wave detector network combining LISA and Taiji can be used to measure the Hubble constant with an uncertainty less than 0.5% in ten years, compared with the network of the ground based gravitational wave detectors which can measure the Hubble constant within a 2% uncertainty in the next five years by the standard siren method. Taiji is a Chinese space borne gravitational wave detection mission planned for launch in the early 2030 s. The pilot satellite mission Taiji-1 has been launched in August 2019 to verify the feasibility of Taiji. The results of a few technologies tested on Taiji-1 are presented in this paper.

In the present work, we consider the cosmological constant model ∝ α −6 , which is well motivated from three independent approaches. As is well known, the hint of varying fine structure constant α was found in 1998. If ∝ α −6 is right, it means that the cosmological constant should also be varying. Here, we try to develop a suitable framework to model this varying cosmological constant ∝ α −6 , in which we view it from an interacting vacuum energy perspective. Then we consider the observational constraints on these models by using the 293 α/α data from the absorption systems in the spectra of distant quasars. We find that the model parameters can be tightly constrained to the very narrow ranges of O(10 −5 ) typically. On the other hand, we can also view the varying cosmological constant model ∝ α −6 from another perspective, namely it can be equivalent to a model containing "dark energy" and "warm dark matter", but there is no interaction between them. We find that this is also fully consistent with the observational constraints on warm dark matter.

We present a new approach to find exact solutions for cosmological models. By requiring the existence of a symmetry transformation vector for the equations of motion of the given cosmolog-ical model (without using either Lagrangian or Hamiltonian), one can find corresponding Hojman conserved quantities. With the help of these conserved quantities, the analysis of the cosmological model can be simplified. In the case of quintessence scalar-tensor models, we show that the Hojman conserved quantities exist for a wide range of V (φ)-potentials and allow to find exact solutions for the cosmic scale factor and the scalar field. Finally, we investigate the general cosmological behavior of solutions by adopting a phase-space view.

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