When we observe a quasar with redshift \(z=0.75\), how far into its past are we looking? If we could see that quasar as it really is right now (that is, if the light from the quasar could somehow reach us instantaneously), would it still look like a quasar? Explain why or why not.

Quasars are some of the most luminous and energetic objects in the universe. They are a type of active galactic nucleus (AGN), typically found at the center of a galaxy surrounded by a gaseous accretion disk. Massive black holes at the centers of these galaxies draw in the surrounding gas and dust, heating it to extreme temperatures and causing it to emit brilliant light and other forms of radiation.

Due to their intense brightness, quasars can be seen over vast distances, making them valuable cosmic beacons for astronomers studying the early universe. However, there's a catch: because light travels at a finite speed, the image we see of a quasar is how it existed in the past, not as it is in the present. If a quasar with a redshift of 0.75 were observed, we would be looking at it as it was billions of years ago.

The relativistic Doppler effect is a phenomenon that occurs when an object emitting light moves at a significant fraction of the speed of light relative to an observer. The classical Doppler effect explains how the pitch of a sound changes as the source moves towards or away from us, but light requires relativistic considerations due to its constant speed.

In the context of astrophysics, the relativistic Doppler effect causes the wavelengths of light emitted by fast-moving objects, like quasars, to stretch or compress. When objects are moving away from us in the expanding universe, their light is redshifted, meaning we see the wavelengths as longer and redder than they were when originally emitted. The observed redshift of a quasar, like the one with a value of 0.75 in the problem, is a key piece of information used to calculate the velocity at which the object is moving away from Earth.

Hubble's Law is a cornerstone of cosmological understanding, asserting that the universe is expanding and that the farther away a galaxy is, the faster it moves away from us. This theory is crucial in interpreting the redshift of distant astronomical objects. The law is typically expressed in a simple equation: \( v = H_0 \times d \) where \( v \) is the galaxy's recession velocity, \( H_0 \) is the Hubble constant (the rate of expansion per unit distance), and \( d \) is the distance from the galaxy to the observer.

Using Hubble's Law scholars can determine the distance to faraway objects like quasars by measuring their redshift. Combining this information with the speed of light allows astronomers to estimate how long ago the light left the quasar, effectively looking back in time.

The ΛCDM (Lambda Cold Dark Matter) model is the prevailing cosmological model for the universe, which includes the Big Bang theory and cosmic inflation. It describes a universe that is composed of normal matter, cold dark matter, and dark energy, characterized by the cosmological constant denoted by 'Λ'.

The ΛCDM model provides the framework for understanding the evolution of the universe from its earliest moments to the present. It allows scientists to make precise predictions about the structure and distribution of galaxies, cosmic microwave background radiation, and the large-scale structure observed in the universe. Calculations related to quasars, including their estimated time delay and their possible appearance today, are often made within the context of the ΛCDM model.

The speed of light, denoted by \( c \) and approximately \( 300,000 \) km/s, is one of the fundamental constants of nature. It represents the maximum speed at which all conventional matter and information in the universe can travel. Moreover, it is not just a limit on speed but also a critical component in the equations of space and time.

In the context of studying quasars and their redshift, the speed of light is what determines the 'lookback time' – the time it takes for light to travel from the quasar to Earth. Understanding that no information can travel faster than the speed of light, it's inferred that when looking at a quasar with a given redshift, we are seeing it as it existed in the past, when the light we now detect was first emitted.