Category: cosmology

Cosmological Metric & Expansion As per the law of conservation of energy and the law of conservation of mass, the total amount of energy including the mass (E=mc2) remains unchanged throughout every step in any process in the universe. The expansion of the universe itself consumes energy which maybe from the stretching of wavelength of photons (Cosmological Redshift), Dark Energy Interactions, etc. To speed up the survey of more than 26,000 galaxies, Stephen A. Shectman designed an instrument capable of measuring 112 galaxies simultaneously. In a metal plate, holes that corresponded to the positions of the galaxies in the sky were drilled. Fiber-optic cables carried the light from each galaxy down to a separate channel on a spectrograph at the 2.5-meter du Pont telescope at the Carnegie Observatories on Cerro Las Campanas in Chile. For maximum efficiency, a specialized technique known as the Drift-Scan Photometry was used, in which the telescope was pointed at the beginning of a survey field and then automated drive was turned off. The telescope stood still as the sky drifted past. Computers read information from the CCD Detector at the same rate as the rotation of the earth, producing one long, continuous image at a constant celestial latitude. Completing the photometry took a total of 450 hours. Different forms of noise exist and their mathematical modelling is different depending upon its properties. Various physical processes evolve the power spectrum of the universe on a large scale. The initial power spectrum imparted due to the quantum fluctuations follows a negative third power of frequency which is a form of Pink Noise Spectrum in three dimensions. The Metric In cosmology, one must first have a definition of space. A metric is a mathematical expression describing points in space. The observation of the sky is done in a spherical geometry; hence a spherical coordinate system shall be used. The distance between two closely spaced points is given by − $$ds^2 = dr^2 + r^2theta ^2 + r^2 sin^2theta dphi^2$$ The following image shows Geometry in the 3-dimensional non-expanding Euclidean space. This geometry is still in the 3-dimensional non-expanding Euclidean space. Hence, the reference grid defining the frame itself would be expanding. The following image depicts the increased metric. A scale factor is put into the equation of the non-expanding space, called the ‘scale factor’ which incorporates expansion of the universe with respect to time. $$ds^2 = a^2(t)left [ dr^2 + r^2theta^2 + r^2 sin^2theta dphi^2 right ]$$ where a(t) is the scale factor, sometimes written as R(t). Whereas, a(t) > 1 means magnification of the metric, while a(t) < 1 means shrinking of the metric and a(t) = 1 means constant metric. As a convention, a(t0) = 1. Comoving Coordinate System In a Comoving Coordinate System, the measuring scale expands along with the frame (expanding universe). Here, the $left [ dr^2 + r^2theta^2 + r^2 sin^2theta dphi^2 right ]$ is the Comoving Distance, and the $ds^2$ is the Proper distance. The proper distance will correspond to the actual distance as measured of a distant galaxy from earth at the time of observation, a.k.a. instantaneous distance of objects. This is because the distance travelled by a photon when it reaches the observer from a distant source will be the one received at $t=t_0$ of the observer, which would mean that the instantaneous observed distance will be the proper distance, and one can predict future distances using the rate factor and the initial measured length as a reference. The concept of Comoving and proper distance is important in measuring the actual value of the number density of galaxies in the given volume of the observed space. One must use the Comoving distance to calculate the density at the time of their formation when the observed photon was emitted. That can be obtained once the rate of expansion of the universe can be estimated. To estimate the rate of expansion, one can observe the change in the distance of an observed distant galaxy over a long period of time. Points to Remember A metric is a mathematical expression describing the points in space. The scale factor determines whether the universe is contracting or expanding. In a comoving coordinate system, the measuring scale expands along with the frame (expanding universe). Proper distance is the instantaneous distance of objects. Comoving distance is the actual distance of objects. Learning working make money

Cosmology – The Expanding Universe Cosmology is the study of the universe. Tracing back in the time, there were several school of thoughts regarding the origin of the universe. Many scholars believed in the Steady State Theory. As per this theory, the universe was always the same, it had no beginning. While there were a group of people who had faith in the Big Bang Theory. This theory predicts the beginning of the universe. There were evidences of hot left out radiation from the Big Bang, which again supports the model. The Big Bang Theory predicts the abundance of light elements in the universe. Thus, following the famous model of Big Bang, we can state that the universe had a beginning. We are living in an expanding universe. The Hubble Redshift In the early 1900’s, the state of the art telescope, Mt Wilson, a 100-inch telescope, was the biggest telescope then. Hubble was one of the prominent scientists, who worked with that telescope. He discovered there were galaxies outside the Milky Way. Extragalactic Astronomy is only 100 years old. Mt Wilson was the biggest telescope until Palmer Observatory was built which had a 200-inch telescope. Hubble was not the only person observing galaxies outside the Milky Way, Humason helped him. They set out on measuring the spectra of nearby galaxies. They then observed a galactic spectrum was in the visible wavelength range with continuous emission. There were emission and absorption lines on top of the continuum. From these lines, we can make an estimate if the galaxy is moving away from us or towards us. When we get a spectrum, we assume the strongest line is coming from H-α. From literature, the strongest line should occur at 6563 Å, but if the line occurs somewhere around 7000Å, we can easily say it is redshifted. From the Special Theory of Relativity, $$1 + z = sqrt{frac{1+frac{v}{c}}{1-frac{v}{c}}}$$ where, Z is the redshift, a dimensionless number and v is the recession velocity. $$frac{lambda_{obs}}{lambda_{rest}} = 1 + z$$ Hubble and Humason listed down 22 Galaxies in their paper. Nearly all these galaxies exhibited redshift. They plotted the velocity (km/s) vs distance (Mpc). They observed a linear trend and Hubble put forward his famous law as follows. $$v_r = H_o d$$ This is the Hubble Redshift Distance Relationship. The subscript r indicates expansion is in the radial direction. While, $v_r$ is the receding velocity, $H_o$ is the Hubble parameter, d is the distance of the galaxy from us. They concluded far away galaxies recede faster from us, if the rate of expansion for the universe is uniform. The Expansion Everything is moving away from us. The galaxies are not stationary, there is some expansion harmonic always. The units of the Hubble parameter are km sec−1Mpc−1. If one goes out a distance of – 1 Mpc, galaxies would be moving at the rate of 200 kms/sec. The Hubble parameter gives us the rate of expansion. As per Hubble and Humason, the value of $H_o$ is 200 kms/sec/Mpc. The data showed all galaxies are moving away from us. Thus, it is apparent that we are at the center of the universe. But Hubble didn’t make this mistake, as per him, in whichever galaxy we live, we would find all other galaxies moving away from us. Thus, the conclusion is that the space between galaxies expand and there is no center of the universe. The expansion is happening everywhere. However, there are some forces that are opposing expansion. Chemical bonds, gravitational force and other attractive forces are holding objects together. Earlier all the objects were close together. Considering the Big Bang as an impulsive force, these objects are set to move away from each other. Time Scale At local scales, Kinematics is governed by Gravity. In the original Hubble’s law, there were some galaxies which showed blue-shift. This can be credited to combined gravitational potential of the galaxies. Gravity has decoupled things from the Hubble’s law. The Andromeda Galaxy is coming towards to us. Gravity is trying to slow things down. Initially the expansion was slowing down, now it is speeding up. There was a Cosmic Jerk because of this. Several estimates to the Hubble parameter has been made. It has evolved over the 90 years from 500 kms/sec/Mpc to 69 kms/sec/Mpc. The disparity in the value was because of the underestimation of distance. The Cepheid Stars were used as distance calibrators, however there are different types of Cepheid stars and this fact was not considered for the estimation of the Hubble parameter. Hubble Time The Hubble constant gives us a realistic estimate of the age of the universe. The $H_o$ would give the age of the universe provided the galaxies have been moving with the same velocity. The inverse of $H_o$ gives us Hubble time. $$t_H = frac{1}{H_o}$$ Replacing the present value of $H_o, t_H$ = 14 billion years. Rate of expansion has been constant throughout the beginning of the Universe. Even if this is not true, $H_o$ gives a useful limit on the age of the universe. Assuming a constant rate of expansion, when we plot a graph between distance and time, the slope of the graph is given by velocity. In this case, the Hubble time is equal to the actual time. However, if the universe had been expanding faster in the past and slower in the present, the Hubble time gives an upper limit of age of the universe. If the universe was expanding slowly before, and speeding up now, then the Hubble time will give a lower limit on age of the universe. $t_H = t_{age}$ − if rate of expansion is constant. $t_H > t_{age}$ − if universe has expanded faster in the past and slower in the present. $t_H < t_{age}$ − if universe has expanded slower in the past and faster in the present. Consider a group of 10 galaxies which are at 200 Mpc from another group of galaxies. The galaxies within a cluster never conclude that the