Category: cosmology

Discuss Cosmology Cosmology is the science and study of the origin, current state and the future of the Universe. This field has been revolutionized by many discoveries made during the past century. This tutorial will attempt at explaining the basic cosmology and summarize the discoveries made in this regard. It gives a deep insight into the Big Bang Theory and the attempts made to discover the Extrasolar Planets. The mysterious CMB (Cosmic Microwave Background Radiation) and its anisotropies which have intrigued astronomers for several years has been explained in detail here. This tutorial will be very useful for budding scientists and astronomers. Learning working make money

Cosmology – Exoplanet Properties First direct image of an extrasolar planet, in 2004, was of a planet of mass 3-10 Mjupiter orbiting around a brown dwarf (2M1207) with a mass of 25 Mjupiter. Techniques like Radial velocity, Transit, Gravitational microlensing, Imaging, Astrometry, etc., has been used for detection of exoplanets. The number of detections have been increasing every year. Until around 2010, the Radial velocity method was used extensively, but now most of the detections are done by the Transit method. There was a spike in the number of detections in 2014, which was when Kepler Space Telescope (KST) started giving the results. A mass-period distribution shows that the Radial velocity method is more biased towards detection of massive planets with a larger period, whereas using the Transit method, planets with lower period are only detected as shown in the following image (Courtesy: NASA Exoplanet Archive). There is a colossal increase in the number of detection of smaller mass planets since the advent of KST. This is evident from the figure given below. The planets detected by KST are divided into two groups: hot massive planets called “Hot Jupiters” and lower mass planets called “Hot Super Earths” (as they are more massive than Earth). When we plot the number of extrasolar planets detected versus distance to them, we find that most of these planets are within 2kpc, which is well within our galaxy. Maybe the planets are not so uncommon in the universe, since our detection is limited to only certain type of planets in a very small part of the universe. Planets are formed from circumstellar disc or proto planetary disc. If planets are formed as a by-product during star formation, maybe the number planets in the universe exceeds the number of stars in universe!! Habitable Zones A Habitable Zone can be defined as the zone around the star where water can exist in its liquid form. Consider a planet at distance $a_p$ from the star as shown in the following figure. A simple method for calculating the temperature of the planet is described as follows. $$left ( frac{L_ast}{4pi a^2_p} right )pi R^2_p(1 – A) = 4pi R^2_p sigma T^4_p$$ and $$frac{L_ast}{4pi R^2_ast} = sigma T^4_ast$$ $$therefore T_p = (1 – A)T_ast sqrt{frac{R_ast}{2a_p}}$$ In our case substituting Lsun = 3.83 x 1026 ap = 1.5 ∗ 1011 and A = 0.3 Will give $T_{Earth} = 255K$. The actual calculation is very involved which includes cloud physics. The Habitable zone in our solar system lies between 0.9 AU and 1.7 AU. The Sun’s luminosity was found to be increasing with time because of the decreasing gas pressure. It was 30% less bright when it started burning hydrogen. This would result is shift of habitable zone away from the Sun. Since Earth is near the inner edge of the Habitable zone, maybe one day it will move out of the zone! Continuously Habitable Zone In short it is termed as CHZ can be defined as the region in which liquid water can exist over the entire Main Sequence lifetime of a star. The KST has detected many extrasolar planets which do lie in the habitable zone. A bio-signature is any substance − such as an element, isotope, molecule, or phenomenon that provides scientific evidence of past or present life. An example is detection of both O2 and CO2 on a planet, which is usually not possible through geological processes alone. This detection is done by analyzing the absorption spectra. Points to Remember Techniques like Radial velocity, Transit, Gravitational microlensing, Imaging, Astrometry, etc., have been used for detection of exoplanets. Radial velocity method is more biased towards detection of massive planets with larger period. Hot massive planets are called “Hot Jupiter’s” and lower mass planets are called “Hot Super Earths”. The number of planets in the universe exceeds the number of stars in the universe. A Habitable zone can be defined as the zone around the star where water can exist in its liquid form. Learning working make money

Cosmology – Useful Resources The following resources contain additional information on Cosmology. Please use them to get more in-depth knowledge on this. Useful Links on Cosmology − Wikipedia Reference for Cosmology Useful Books on Cosmology To enlist your site on this page, please drop an email to [email protected] Learning working make money

Cosmology – Modelling the CMB Anisotropies When we look at the refined, corrected all sky CMB map, there is a lot of foreground contamination, which is a kind of anisotropy in these maps. We can see that these foreground emissions are from the milky way galaxy. The intensity of CMB is high along the plane of galactic plane and reduces in intensity as we move away. In these, we can observe secondary anisotropies, which are synchrotron emissions from the galaxy. These emissions make up the foreground contamination. To look at the CMB emission from the sky, we need to subtract these foreground emissions. The following image shows the CMB with foreground emissions. Dipole Anisotropy There is another kind of anisotropy, which was found in the CMB all sky map, it is called as Dipole Anisotropy. It is not associated with the early universe. This can be represented using spherical harmonic functions. If there is a pattern on spherical surface and we want to map it using mathematical functions, we can do so using trigonometric functions. So, when we map, it can be a monopole – same at every direction, or a dipole – flips properties when rotated by 180 degrees. Similarly, we have quadrupole and so on. For a complex pattern, it can be expressed as the sum of these monopole, dipole, quadrupole, etc. The CMB is modelled in such a manner that one of the major sources of anisotropy in the all sky map is this dipole anisotropy, but it is not primordial modelling of CMB. This can be seen in the image below. The dipole direction we get to see is not any random direction. Dipole anisotropy has a direction. We see the CMB intensity along a specific direction. This direction is due to the solar system velocity vector. The velocity of earth can be represented with respect to sun or centre of galaxy. The direction in which earth is moving, we observe a Blueshift and Redshift and the dipole lies along this direction. The above image has a typical dipole appearance because our Galaxy is moving in a specific direction. The result is – one side of the sky will appear Redshifted and the other side of the sky will appear Blueshifted. In this case, Redshifting means the photons are longer in wavelength = cooler (so backwards from their name, they look blue in the above diagram). We can say, the earth is moving in some specific direction with respect to sun/galactic centre/ CMB in the sky at a given instant. Then, if we look at any angle and measure the temperature for CMB, it would be different. This is because, we are measuring photons which are either Blueshifted or Redshifted and depends on the line of sight of photons in the sky. Points to Remember The foreground contamination in CMB all sky map is called anisotropy of CMB. These emissions are from our own milky way galaxy. The 2 types of anisotropies are: Dipole Anisotropy and Angular Power Spectrum Anisotropy. Dipole anisotropy is in a specific direction, whereas Angular Power Spectrum anisotropy is spread everywhere. Learning working make money

Cosmology – Extrasolar Planet Detection Astrobiology is the study of origin, evolution, distribution and future of life in the universe. It is concerned with discovering and detecting Extrasolar Planets. Astrobiology addresses the following points − How does life begin and evolve? (biology + geology + chemistry + atmospheric sciences) Are there worlds beyond earth that are favorable for life? (astronomy) What would be the future of life on earth? Astronomy addresses the following points − How to detect the planetary system around other stars? One of the method is direct imaging, but it is a very difficult task because planets are extremely faint light sources compared to stars, and what little light comes from them tends to be lost in the glare from their parent star. Contrast is better when the planet is closer to its parent star and hot, so that it emits intense infrared radiation. We can make images in the infrared region. Techniques for Extrasolar Planet Detection The most efficient techniques for extrasolar planet detection are as follows. Each of these are also explained in detail in the subsequent chapters. Radial Velocity Method It is also called as the Doppler method. In this − The star planet system revolves around their barycenter, star wobbles. Wobbling can be detected by Periodic Red/Blue shifts. Astrometry – measuring the objects in the sky very precisely. Transit Method Transit Method (Kepler space telescope) is used to find out the size. The dip in brightness of star by planet is usually very less, unlike in a binary system. Direct Imaging Imaging the planet using a telescope. Let us look at a case study done on Radial Velocity Method. Case Study This case study is on the Circular orbit and the plane of the orbit perpendicular to the plane of the sky. The time taken by both around the barycenter will be same. It will be equal to the time difference between two Redshift or Blueshift. Consider the following image. At A and C – full velocity is measured. At C, velocity is zero. Vrmax = V* is the true velocity of the star. P is the time-period of the star as well as the planet. θ is the phase of orbit. Star Mass – M*, Orbit radius a*, planet mass mp. From center of mass equation, $$m_p a_p = M_ast a_ast$$ From equation of velocity, $$V_ast = frac{2pi a_ast}{P}$$ $$Rightarrow a_ast = frac{PV_ast}{2pi}$$ From Kepler’s Law, $$P^2 = frac{4pi^2a_p^3}{GM_ast}$$ $$Rightarrow a_p = left ( frac{P^2GM_ast}{4pi^2} right)^{1/3}$$ From the above equations, we get − $$Rightarrow m_p = left( frac{P}{2pi G} right)^{1/3}M_ast^{2/3}V_ast$$ We get: $m_p, a_p$ and $a_ast$. The above equation is biased towards most massive planets close to the star. Points to Remember Astrobiology is the study of origin, evolution, distribution and future of life in the universe. Techniques to detect the extrasolar planets are: Radial Velocity Method, Transit Method, Direct Imaging, etc. Wobbling can be detected by periodic red/blue shifts and Astrometry. The Radial Velocity Method is biased towards detecting massive planets close to the star. Learning working make money

CMB Temperature at Decoupling We should first understand what characterizes the decoupling. We know that energies were much higher to such an extent that matter existed only in the form of Ionized Particles. Thus, at decoupling and recombination epochs, the energy had to drop to permit the ionization of hydrogen. An approximate calculation can be made to the estimation of temperature at the time of decoupling. This has been performed as follows − First, consider only the ionization of ground state hydrogen. $$hv approx k_BT$$ $$therefore T approx frac{hv}{k_B}$$ For ionization of the ground state hydrogen, hν is 13.6 eV and kB is the Boltzmann Constant 8.61 × 10−5 eV/K that reveals the temperature to be 1.5 × 105 kelvin. This essentially tells us that if the temperature is below 1.5 × 105 K, the neutral atoms can begin to form. We know that the ratio of photons to baryons is about 5 × 1010. Hence even at the tail of the graph where the number of photons reduces, there will still be sufficient photons to ionize the hydrogen atoms. Moreover, recombination of electron and proton does not guarantee a ground state hydrogen atom. Excited states require lesser energy for ionization. Hence a disciplined statistical analysis should be performed case by case to obtain an accurate value. Computations set the temperature to be around 3000K. For explanations sake, we consider the case of exciting hydrogen into the first excited state. The general expression for the ratio of the number of photons with energy more than ΔE, Nγ (> ΔE) to the total number of photons Nγ is given by − $$frac{N_gamma(> Delta E)}{N_gamma} propto e^{frac{-Delta E}{kT}}$$ For the case of exciting hydrogen to the first excited state, ΔE is 10.2 eV. Now, if we consider a highly conservative number of at least 1 photon with energy more than 10.2 for every baryon (keeping in mind that the ratio is 5 × 1010, we obtain temperature from the equation 3 as 4800 K (Inserted Nγ(> ΔE) = Np). This is the temperature to create a population of neutral hydrogen atoms in the first excited state. The temperature to ionize this is significantly lesser. Thus, we obtain a better estimate than 1.5 × 105 K that is closer to the accepted value of 3000 K. Redshift – Temperature Relationship To understand the relationship between redshift and temperature, we employ the following two methods as described below. Method 1 From Wien’s Law, we know that $$lambda_mT = constant$$ To relate this to the redshift, we use − $$1+z = frac{lambda_0}{lambda_e}$$ As $λ_oT_o = λ_eT(z)$, we get − $$T(z) = T_0frac{lambda_0}{lambda_e} = T_0(1+z)$$ Setting To as the current value 3K, we can get temperature values for a given redshift. Method 2 In terms of frequency, we know − $$v_0 = frac{v_e}{1+z}$$ $$B_vdv = frac{2hv^3}{c^2} frac{dv}{e^{hv/kT}-1}$$ This tells us about the net energy of the photons for an energy interval and hν is the energy of a single photon. Hence, we can obtain the number of photons by Bνdν/hν. If $n_{νo}$ is for present and $n_{νe}$ for emitted, we get − $$frac{n_{v_e}}{n_{v_0}} = (1+z)^3$$ On simplification, we get, $$n_{v_0} =frac{2v_c^2}{c^2}frac{dv_c}{e^{hv/kT}-1}frac{1}{(1+z)^3}=frac{2v_0^2}{c^2}frac{dv_c}{e^{hv/kT}-1}$$ This gives us the Wien’s Law again and thus it can be concluded that − $$T(z) = T_0frac{lambda_0}{lambda_e} = T_0(1+z)$$ Points to Remember The early universe was very hot, ∼ 3000K. Current measurements reveal the universe’s temperature to be close to 3K. The further back we go in time, the temperature increases proportionally. Learning working make money

Cosmology – Quick Guide 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

Cosmology – Cosmic Microwave Background The CMB (Cosmic Microwave Background) essentially is constituted by the photons of the time when matter and radiation was in equilibrium. By the 1920s, the idea of an expanding universe was accepted and could answer several questions. But questions about the abundance of heavier elements and the abundance were left unanswered. Moreover, the expanding universe implied that the density of matter should decrease down to 0. In 1948, George Gammow and Ralph Alpher explained the Origin of heavier elements and the abundance using “Big Bang”. They along with Robert Herman predicted the existence of “Relict Radiation” or radiation remaining from the “Big Bang”. The predicted temperature for this remnant radiation was between 50-6 K. In 1965, Robert Dicke, Jim Peebles and David Wilkinson along with the Amo Perizias’ Research Group experimentally detected the CMB. The early universe was very hot and the energy was too high for the matter to remain neutral. Hence, matter was in the ionized form – Plasma. The Radiation (photons) and Matter (plasma) interacted mainly through the following three processes. Compton Scattering − (Major Interaction Process) Inelastic scattering between high energy photon and low energy charged particle. Thomson Scattering − Elastic scattering of photon by a free charged particle. Inverse Compton Scattering − High energy charged particle and low energy photon. These interactions finally resulted in matter and radiation being in Thermal equilibrium. Thermal Equilibrium In Thermal equilibrium, the radiation obeys the Planck Distribution of Energy, $$B_v(T) = frac{2hv^3}{c(e^{hv/k_BT}-1)}$$ During this time, due to the highly frequent interactions, the mean free path of photons was very small. The universe was Opaque to radiation. The early universe was radiation dominated. The universe evolved in such a way that matter and radiation reached Thermal Equilibrium and their energy density became equal. This can be seen from the graph showing evolution of density with the scale factor. Let us find out the scale factor (time) (a(t)) at which the matter and the radiation reached equilibrium. $$rho_m propto frac{1}{a^3}, : rho_r propto frac{1}{a^4}$$ $$frac{rho_{m,t}}{rho_{r,t}} = frac{Omega_{m,t}}{Omega_{r,t}} = frac{Omega_{m,0}}{Omega_{r,0}}a(t)$$ At equilibrium, $$frac{rho_{m,t}}{rho_{r,t}} = frac{Omega_{m,t}}{Omega_{r,t}} = 1$$ $$Rightarrow frac{Omega_{m,0}}{Omega_{r,0}}a(t) = 1 : Rightarrow a(t) = 2.96 times 10^{-4}$$ using $Omega_{m,0} = 0.27$ and $Omega_{r,0} = 8 times 10^{−5}$. The red shift corresponding to this scale factor is given by − $$z = 1/a(t)-1 approx 3375$$ The energy density of the radiation went down due to the expansion of the universe. Thus the universe started to cool down. As the energy of the photons started to decrease, neutral atoms started to form. Thus, around a redshift of 1300, neutral Hydrogen started to form. This era had a temperature close to 3000K. The interaction between matter and radiation became very infrequent and thus the universe started becoming transparent to radiation. This time-period is called “Surface of last scattering” as the mean free path of the photons became very large due to which hardly any scattering took place after this period. It is also called as “Cosmic Photosphere”. Points to Remember CMB is constituted by the photons of the time when matter and radiation was in equilibrium. The early universe was very hot and the energy was too high for matter to remain neutral, so it existed as ionized matter-Plasma. Compton Scattering, Thomson Scattering, Inverse Compton Scattering were the 3 matter-radiation interaction processes then. Universe evolved such that matter and radiation reached Thermal equilibrium. Learning working make money

Cosmology – Type 1A Supernovae For any given redshift (z), we have two values for the distance − Angular Diameter Distance (dA) Luminosity Distance (dL) There is no unique definition of “cosmological” distance in the universe. The choice of the distance depends on the purpose and convenience of application. To test the predicted trend of how angular size of an object varies with redshift, a standard size yardstick is needed in the sky. This should be an object which − is very luminous, so that it can be detected at z > 1. is very large, so that we can resolve its angular size. has not morphologically evolved over cosmologically significant time (z ∼ 1 corresponds to a look back time of about 7 Gyr). Some objects (like cD galaxies) satisfy the first two criteria. But almost every object is found to morphologically evolve with time. In general, astrophysical objects (extended sources) tend to be intrinsically smaller in the past because they are still forming. Luminosity Distance Luminosity distance depends on cosmology. The dependence of luminosity distance on cosmology makes it a useful measure of the cosmological parameters. The cosmological parameters can be estimated if we can find a standard candle that does not intrinsically evolve and exists from the local to the high redshift universe. A standard candle is one which does not differ in its luminosity from source to source. The premise is that any difference in the estimated luminosity of standard candles has to be due to cosmology. One such candle is Type Ia Supernovae. Type 1a Supernovae (SNe) These are the result of explosion of a white dwarf after sufficient mass accretion from its companion, a red giant or similar main sequence star, in a binary system. After the red giant comes closer than the Roche lobe distance of the White dwarf, the mass transfer begins and eventually the white dwarf explodes giving out huge amount of energy, leaving no core behind. These are called Type 1a Supernovae. The typical rate of Type 1a Supernovae explosion in a galaxy is 1 per century. The search for Type 1a SNe has been going on with different teams − High z Supernova Search Team (Brian Schmidt, Adam Reiss et al.) Supernova Cosmology Project (Saul Perlmutter et al.) There has been another research team called Carnegie Supernovae Project who has given similar results. The similarity of results from different teams show the cosmological nature of Type 1a SNe. Hence, they are efficient standard candles. Points to Remember There is no unique definition of “cosmological” distance in the universe. Angular Diameter Distance and Luminosity Distance are the most used. A standard candle is the one which does not differ in its luminosity from source to source. Type 1a SNe satisfies the criteria of being a standard candle. Learning working make money

Cosmology – Luminosity Distance As discussed in the previous chapter, the angular diameter distance to a source at red shift z is given by − $$d_wedge (z_{gal}) = frac{c}{1+z_{gal}}int_{0}^{z_{gal}} frac{1}{H(z)}dz$$ $$d_wedge(z_{gal}) = frac{r_c}{1+z_{gal}}$$ where $r_c$ is comoving distance. The Luminosity Distance depends on cosmology and it is defined as the distance at which the observed flux f is from an object. If the intrinsic luminosity $d_L$ of a distant object is known, we can calculate its luminosity by measuring the flux $f$ which is determined by − $$d_L(z) = sqrt{frac{L}{4pi f}}$$ The Photon Energy gets red shifted. $$frac{lambda_{obs}}{lambda_{emi}} = frac{a_0}{a_e}$$ where $lambda_{obs}, lambda_{emi}$ are observed and emitted wave lengths and $a_0, a_e$ are corresponding scale factors. $$frac{Delta t_{obs}}{Delta t_{emi}} = frac{a_0}{a_e}$$ where $Delta_t{obs}$ is observed as the photon time interval, while $Delta_t{emi}$ is the time interval at which they are emitted. $$L_{emi} = frac{nhv_{emi}}{Delta t_{emi}}$$ $$L_{obs} = frac{nhv_{obs}}{Delta t_{obs}}$$ $Delta t_{obs}$ will take more time than $Delta t_{emi}$ because the detector should receive all the photons. $$L_{obs} = L_{emi}left ( frac{a_0}{a_e} right )^2$$ $$L_{obs} < L_{emi}$$ $$f_{obs} = frac{L_{obs}}{4pi d_L^2}$$ For a non-expanding universe, luminosity distance is same as the comoving distance. $$d_L = r_c$$ $$Rightarrow f_{obs} = frac{L_{obs}}{4pi r_c^2}$$ $$f_{obs} = frac{L_{emi}}{4 pi r_c^2}left ( frac{a_e}{a_0} right )^2$$ $$Rightarrow d_L = r_cleft ( frac{a_0}{a_e} right )$$ We are finding luminosity distance $d_L$ for calculating luminosity of emitting object $L_{emi}$ − Interpretation − If we know the red shift z of any galaxy, we can find out $d_A$ and from that we can calculate $r_c$. This is used to find out $d_L$. If $d_L ! = r_c(a_0/a_e)$, then we can’t find Lemi from $f_{obs}$. The relation between Luminosity Distance $d_L$ and Angular Diameter Distance $d_A.$ We know that − $$d_A(z_{gal}) = frac{d_L}{1+z_{gal}}left ( frac{a_0}{a_e} right )$$ $$d_L = (1 + z_{gal})d_A(z_{gal})left ( frac{a_0}{a_e} right )$$ Scale factor when photons are emitted is given by − $$a_e = frac{1}{(1+z_{gal})}$$ Scale factor for the present universe is − $$a_0 = 1$$ $$d_L = (1 + z_{gal})^2d_wedge(z_{gal})$$ Which one to choose either $d_L$ or $d_A$? For a galaxy of known size and red shift for calculating how big it is, then $d_A$ is used. If there is a galaxy of a given apparent magnitude, then to find out how big it is, $d_L$ is used. Example − If it is given that two galaxies of equal red shift (z = 1) and in the plane of sky they are separated by 2.3 arc sec then what is the maximum physical separation between those two? For this, use $d_A$ as follows − $$d_A(z_{gal}) = frac{c}{1+z_{gal}}int_{0}^{z_{gal}} frac{1}{H(z)}dz$$ where z = 1 substitutes H(z) based on the cosmological parameters of the galaxies. Points to Remember Luminosity distance depends on cosmology. If the intrinsic luminosity $d_L$ of a distant object is known, we can calculate its luminosity by measuring the flux f. For a non-expanding universe, luminosity distance is same as the comoving distance. Luminosity distance is always greater than the Angular Diameter Distance. Learning working make money