A quantity without physical units that characterizes a system's behavior, often used in scaling laws.A dimensionless parameter is a quantity that has no physical units, meaning it is a pure number. Such parameters often emerge from ratios of physical quantities that cancel out their dimensions, and they play a crucial role in characterizing the behavior of systems across different scales. For example, the Reynolds number in fluid dynamics, which compares inertial forces to viscous forces, determines whether flow is laminar or turbulent, regardless of the specific fluid or pipe size. Dimensionless parameters are the backbone of scaling laws, allowing scientists to apply results from small-scale models to large-scale phenomena, from laboratory experiments to astrophysical objects.
The systematic use of dimensionless parameters in physics and engineering dates back to the late 19th and early 20th centuries, with figures like Osborne Reynolds (Reynolds number, 1883) and Ludwig Prandtl (Prandtl number, early 1900s). The formal theory of dimensional analysis was developed by Edgar Buckingham in 1914 with his π theorem.
Dimensionless parameters arise when you combine physical quantities in a way that their units cancel. For instance, in the context of the articles listed, a parameter like the ratio of a system's size to some characteristic length scale can be dimensionless. These parameters often represent the relative importance of competing effects—such as gravity versus surface tension, or quantum effects versus thermal fluctuations. By varying a single dimensionless parameter, one can explore a whole family of systems that behave identically when scaled appropriately, a principle known as similarity or self-similarity. This is why they are so powerful: they reduce the complexity of a problem by collapsing many variables into a few key numbers.
Dimensionless parameters are used in wind tunnel testing of aircraft, in modeling ocean currents and atmospheric circulation, in designing chemical reactors, and in understanding phase transitions in condensed matter physics, as hinted by the titles about scaling and critical phenomena.
The fine-structure constant, a dimensionless number approximately equal to 1/137, is so fundamental that it determines the strength of electromagnetic interactions between charged particles, and its value cannot be predicted by current theory—it must be measured experimentally.
| \kappa'(\theta) | variable | transformed convergence — convergence |
| \kappa(\theta) | variable | original convergence — convergence |
| \lambda | variable | transformation parameter — dimensionless parameter |
| alpha_parallel | variable | BAO scale distortion parameter along line-of-sight — dimensionless parameter |
| H_fid | variable | fiducial Hubble parameter — hubble parameter metre per second per metre |
| r_d_fid | variable | fiducial sound horizon at drag epoch — length metre |
| H | variable | true Hubble parameter — hubble parameter metre per second per metre |
| r_d | variable | true sound horizon at drag epoch — length metre |
