In the theory of general relativity, matter and energy can bend space-time and disturb it’s shape. So if an astrophysical or cosmic process involves periodic or sudden movement of a large amount of matter, in a non-symmetric manner, it can produce gravitational waves. There are many such processes: Binary mergers consisted of black hole and/or neutron stars, supernovae and fast spinning objects are considered astrophysical in nature. Gravitational waves from inflation, phase transitions in early universe, creation of primordial black holes and cosmic string are considered cosmic in nature.

The relationship between matter-energy and space-time is mutual; if space-time bends in a certain manner in a region, matter in that region will follow geodesics altered by that certain manner of bending. We can use this ti detect gravitational waves. Although according the Equivalence principle one can not measure the change in the gravitational field locally (at a single point), we can build a detector that is spread in space. There are resonant-mass detectors like Weber bar and laser interferometers like *LIGO* and *VIRGO*. We can also look at natural phenomena to find a trace of gravitational waves; by measuring for example *Pulsar Timing Arrays*, astrometry of distant Quasars, etc…

Right now, our instruments can only detect the most drastic (and close in terms of cosmic scales) astrophysical events. Once they surpass a certain sensitivity, a new window opens and we can suddenly detect gravitational waves from hundreds or thousands of weaker events that are happening simultaneously. In this case, we can no longer resolve individual events and the signal we will receive is called the Stochastic Gravitational Wave Background (SGWB). At first, this signal might seem very random but we can measure it’s statistical properties and learn a great deal about the types of sources that make up the signal and their distributions. It is worth mentioning that some of the cosmic processes of early universe are expected to produce gravitational waves that are stochastic in nature (are not a result of superposing the signal of many individual events).

There are many ways to study SGWB; simple statistical analyses, multifractal analysis, machine learning, etc… One of the interesting ways to do so, is with the help of Topological Data Analysis (TDA). It is based on the well tested assumption that we can map gravitational wave data, whether it be 1+1 dimensional time series or 2+1 dimensional sky maps, to a topological space and measure it’s geometrical and topological properties. In other words, TDA tries to determine the shape and structure of data. It has been shown that TDA methods are very resistant to noise and also they can measure some properties that other methods of data analysis are blind to.

My name is Ali Salehi and I am a MSc student at Shahid Behesti university. In this project, I will first try to simulate gravitational wave signals created as a result of Primordial Black Hole (PBH) formation in the early universe; then I’ll use TDA to find measurables with which we can characterize an SGWB constituted of those signals. The main goal of the project is to first find the best methods to detect the presence of PBHs and then to figure out how much of an impact different PBH formation scenarios have on SGWB. Since PBHs are one of the candidates for dark matter, measuring their abundance (it can be zero, meaning they might not exist at all) and other properties can be a big step towards understanding cosmology. For a comprehensive report on the most recent constraints on primordial black holes, look at arXiv:2002.12778.