Category: cosmology

Hubble Parameter & Scale Factor In this chapter, we will discuss regarding the Hubble Parameter as well as the Scale Factor. Prerequisite − Cosmological Redshift, Cosmological Principles. Assumption − The universe is homogenous and isotropic. Hubble’s Constant with Fractional Rate of Change of Scale Factor In this section, we will relate the Hubble’s Constant with fractional rate of Change of Scale Factor. We can write velocity in the following manner and simplify. $$v = frac{mathrm{d} r_p}{mathrm{d} t}$$ $$= frac{d[a(t)r_c}{dt}$$ $$v = frac{mathrm{d} a}{mathrm{d} t} ast frac{1}{a} ast (ar_c)$$ $$v = frac{mathrm{d} a}{mathrm{d} t} ast frac{1}{a} ast r_p$$ Here, v is the recessional velocity, a is the scale factor and rp is the proper distance between the galaxies. Hubble’s Empirical Formula was of the nature − $$v = H ast r_p$$ Thus, comparing the above two equations we obtain − Hubble’s Parameter = Fractional rate of change of the scale factor $$H = da/dt ast 1/a$$ Note − This is not a constant since the scale factor is a function of time. Hence it is called the Hubble’s parameter and not the Hubble’s constant. Empirically we write − $$H = V/D$$ Thus, from this equation, we can infer that since D is increasing and V is a constant, then H reduces with the time and expansion of the universe. Friedmann Equation in Conjunction with the Robertson-Walker Model In this section, we will understand how the Friedmann Equation is used in conjunction with the Robertson-Walker model. To understand this, let us take the following image which has a test mass at distance rp from body of mass M as an example. Taking into consideration the above image, we can express force as − $$F = G ast M ast frac{m}{r^2_p}$$ Here, G is the universal gravitational constant and ρ is the matter density inside the observable universe. Now, assuming uniform mass density within the sphere we can write − $$M = frac{4}{3} ast pi ast r_p^3 ast rho$$ Using these back in our force equation we get − $$F = frac{4}{3} ast pi ast G ast r_p ast rho ast m$$ Thus, we can write the potential energy and kinetic energy of the mass m as − $$V = -frac{4}{3} ast pi ast G ast r^2_p ast m ast rho$$ $$K.E = frac{1}{2} ast m ast frac{mathrm{d} r_p^2}{mathrm{d} t}$$ Using the Virial Theorem − $$U = K.E + V$$ $$U = frac{1}{2} ast m ast left ( frac{mathrm{d} r_p}{mathrm{d} t} right )^2 – frac{4}{3} ast pi ast G ast r_p^2 ast m ast rho$$ But here, $r_p = ar_c$. So, we get − $$U = frac{1}{2} ast m ast left ( frac{mathrm{d} a}{mathrm{d} t} right )^2 r_c^2 – frac{4}{3} ast pi ast G ast r_p^2 ast m ast rho$$ On further simplification, we obtain the Friedmann equation, $$left ( frac{dot{a}}{a} right )^2 = frac{8pi}{3} ast G ast rho + frac{2U}{m} ast r_c^2 ast a^2$$ Here U is a constant. We also note that the universe we live in at present is dominated by matter, while the radiation energy density is very low. Points to Remember The Hubble parameter reduces with time and expansion of the universe. The universe we live in at present is dominated by matter and radiation energy density is very low. Learning working make money

Cosmology – Spiral Galaxy Rotation Curves In this chapter, we will discuss regarding Spiral Galaxy Rotation Curves and evidence for Dark Matter. Dark Matter and Observational Fact about Dark Matter The Early Evidence of Dark Matter was the study of the Kinematics of Spiral Galaxy. The Sun is offset 30,000 lightyears from the centre of our Galaxy. The galactic centric velocity is 220 km/s. Why is the velocity 220 km/s not 100 km/s or 500 km/s? What governs the circular motion of object? Mass enclosed within the radius helps to detect the velocity in the Universe. Rotation of Milky Way or Spiral Galaxy – Differential Rotation Angular Velocity varies with the distance from the centre. Orbital time-period depends on the distance from the centre. Material closer to Galactic centre has a shorter time-period and material far away from the Galactic Centre has a larger time period. Rotation Curve Predict the velocity change with the Galactic centric radius. The curve which gives the velocity changes with the orbital radius. When we see things move, we think that it is gravity which influences the rotation. Mass distribution varies with the radius. Matter density will predict the rotation curve. The rotation curve based on the matter density, which varies with radius. Surface Brightness We choose the patch and see how much of light is coming out. Amount of the light coming from the patch is called as the Surface Brightness. Its unit is mag/arcsec2. If we find the surface brightness varies with the radius, we can find the luminous matter varies with radius. $$mu(r) propto exp left( frac{-r}{h_R} right )$$ $h_R$ is scale length. $mu(r) = mu_o ast exp left( frac{-r}{h_R} right )$ $h_R$ is nearly 3 kpc for the Milky way. Spiral Galaxies For the Astronomers to understand the rotational curve, they split the Galaxies into two components, which are − Disk Bulge The following image shows a Central spherical bulge + Circular disk. Stellar and gas distribution is different in the bulge and the disk. Kinematics of Spiral Galaxies The Circular velocity of any object – For the bulge is (r < Rb). $$V^2(r) = G ast frac{M(r)}{r}$$ $$M(r) = frac{4pi r^3}{3} ast rho_b$$ For the disk – (Rb < r < Rd) $$V^2(r) = G ast frac{M(r)}{r}$$ Bulge has a roughly constant density of stars. Density within the Bulge is constant (not changing with the distance within the bulge). In a disk, the stellar density declines with the radius. The radius increases then the luminous matter decrease. In Bulk – $V(r) propto r$ In Disc – $V(r) propto 1/sqrt{r}$ Rotational Curve of Spiral Galaxies Through the Spectroscopy (nearby galaxies – spatially resolved the galaxy), we produce the rotation curve. As mentioned above, we see that the rotation curve is flat at the outer regions, i.e. the things are moving fast in the outer regions, which is generally not expected to be in this form. The orbital speed increases with the increase in the radius of the inner region, but it flattens in the outer region. Dark Matter The Dark Matter is said to be the Non-Luminous Component of the Universe. Let us understand about dark matter through the following pointers. The flat rotation curves are counter to what we see for the distribution of stars and gas in the spiral galaxies. The surface luminosity of the disk falls off exponentially with radius, implying that the mass of luminous matter, mostly stars, is concentrated around the galactic center. The flattening of the rotation curve suggests that the total mass of the galaxy within some radius r is increasing always with increase in r. This can only be explained if there is a large amount of invisible gravitating mass in these galaxies which is not giving out electromagnetic radiation. The rotation curve measurements of spiral galaxies is one of the most compelling set of evidences for dark matter. Evidence of Dark Matter Missing Mass – 10 times the luminous mass. Most of this dark matter must be in the halo of the galaxy: Large amounts of dark matter in the disk can disturb the long-term stability of the disk against tidal forces. Some small fraction of the dark matter in the disk can be baryonic – dim stars (brown dwarfs, black dwarfs), and compact stellar remnants (neutron stars, black holes). But such baryonic dark matter cannot explain the full scale of missing mass in galaxies. Density Profile of Dark Matter – $M(r) propto r$ and $rho(r) propto r^{−2}$. The rotation curve data for spiral galaxies are consistent with the dark matter distributed in their halo. This dark halo constitutes much of the galaxy’s total mass. All baryonic matter (stars, star clusters, ISM, etc.) are held together by the gravitational potential of this dark matter halo. Conclusion Dark matter has only been detected through their gravitational interaction with an ordinary matter. No interaction with light (no electromagnetic force) has yet been observed. Neutrinos − Charge less, weakly interacting, but mass is too less (< 0.23 eV). DM particles should have E > 10 eV or so to explain structure formation. Weakly Interacting Massive Particles (WIMPS) can be the source of Dark Matter. Points to Remember Material closer to Galactic centre has a shorter time-period. Bulge has a roughly constant density of stars. Surface luminosity of the disk falls off exponentially with radius. Large amounts of dark matter in the disk can disturb the long-term stability of the disk against tidal forces. Learning working make money

Cosmology – Cepheid Variables For a very long time, nobody considered galaxies to be present outside our Milky Way. In 1924, Edwin Hubble detected Cepheid’s in the Andromeda Nebula and estimated their distance. He concluded that these “Spiral Nebulae” were in fact other galaxies and not a part of our Milky Way. Hence, he established that M31 (Andromeda Galaxy) is an Island Universe. This was the birth of Extragalactic Astronomy. Cepheid’s show a periodic dip in their brightness. Observations show that the period between successive dips called the period of pulsations is related to luminosity. So, they can be used as distance indicators. The main sequence stars like the Sun are in Hydrostatic Equilibrium and they burn hydrogen in their core. After hydrogen is fully burned, the stars move towards the Red Giant phase and try to regain their equilibrium. Cepheid Stars are post Main Sequence stars that are transiting from the Main Sequence stars to the Red Giants. Classification of Cepheids There are 3 broad classes of these pulsating variable stars − Type-I Cepheids (or Classical Cepheids) − period of 30-100 days. Type-II Cepheids (or W Virginis Stars) − period of 1-50 days. RR Lyrae Stars − period of 0.1-1 day. At that time, Hubble was not aware of this classification of variable stars. That is why there was an overestimation of the Hubble constant, because of which he estimated a lower age of our universe. So, the recession velocity was also overestimated. In Cepheid’s, the disturbances propagate radially outward from the centre of the star till the new equilibrium is attained. Relation between Brightness and Pulsation Period Let us now try to understand the physical basis of the fact that higher pulsation period implies more brightness. Consider a star of luminosity L and mass M. We know that − $$L propto M^alpha$$ where α = 3 to 4 for low mass stars. From the Stefan Boltzmann Law, we know that − $$L propto R^2 T^4$$ If R is the radius and $c_s$ is the speed of sound, then the period of pulsation P can be written as − $$P = R/c_s$$ But the speed of sound through any medium can be expressed in terms of temperature as − $$c_s = sqrt{frac{gamma P}{rho}}$$ Here, γ is 1 for isothermal cases. For an ideal gas, P = nkT, where k is the Boltzmann Constant. So, we can write − $$P = frac{rho kT}{m}$$ where $rho$ is the density and m is the mass of a proton. Therefore, period is given by − $$P cong frac{Rm^{frac{1}{2}}}{(kT)^{{frac{1}{2}}}}$$ Virial Theorem states that for a stable, self-gravitating, spherical distribution of equal mass objects (like stars, galaxies), the total kinetic energy k of the object is equal to minus half the total gravitational potential energy u, i.e., $$u = -2k$$ Let us assume that virial theorem holds true for these variable stars. If we consider a proton right on the surface of the star, then from the virial theorem we can say − $$frac{GMm}{R} = mv^2$$ From Maxwell distribution, $$v = sqrt{frac{3kT}{2}}$$ Therefore, period − $$P sim frac{RR^{frac{1}{2}}}{(GM)^{frac{1}{2}}}$$ which implies $$P propto frac{R^{frac{3}{2}}}{M^{frac{1}{2}}}$$ We know that – $M propto L^{1/alpha}$ Also $R propto L^{1/2}$ So, for β > 0, we finally get – $P propto L^beta$ Points to Remember Cepheid Stars are post Main Sequence stars that are transiting from the Main Sequence stars to Red Giants. Cepheid’s are of 3 types: Type-I, Type-II, RR-Lyrae in decreasing order of pulsating period. Pulsating period of Cepheid is directly proportional to its brightness (luminosity). Learning working make money

Cosmology – Dark Energy The area of Dark Energy is a very grey area in astronomy because it is a free parameter in all equations, but there is no clear idea what exactly this is. We will start with the Friedmann’s equations, $$left ( frac{dot{a}}{a} right)^2 = frac{8pi G}{3}rho – frac{k ast c^2}{a^2}$$ Most of the elementary books on cosmology, they all start with describing the dark energy from this episode that before Hubble’s observation, the universe is closed and static. Now, for the universe to be static in the right side, both the terms should match and they should be zero, but if the first term is greater than the second term, then the universe will not be static, so Einstein dropped the free parameter ∧ into the field equation to make the universe static, so he argued that no matter what the first term is compared to the second term, you can always get a static universe if there is one more component in the equation, which can compensate the dis-match between these two terms. $$left ( frac{dot{a}}{a} right )^2 = frac{8pi G}{3}rho – frac{k ast c^2}{a^2} + frac{wedge}{3}$$ $$left ( frac{ddot{a}}{a} right ) = -frac{4 pi G}{3}left ( rho + frac{3P}{c^2} right ) + frac{wedge}{3}$$ Where $P = rho ast c^2/3$ and $wedge = rho ast c^2$ is the Cosmological Parameter. (Negative sign is only because of attraction) In the above equation (acceleration equation) − $3P/c^2$ is the negative pressure due to radiation, $-4pi G/3$ is the attraction due to gravity, and $wedge/3$ makes a positive contribution. The third term acts as a repulsive force because another part of the equation is attractive. The physical significance of the equation is that ˙a = 0 because there was not any evidence which shows that the universe is expanding. What if these two terms are not matching with each other, so it is better to add a component and depending on the offset we can always change the value of the free parameter. That time there was no physical explanation about this cosmological parameters, which is why when the explanation of the expanding universe was discovered in 1920s, where Einstein immediately had to throw this constant out. The explanation of this cosmological constant is still in use because it explains a different version of the universe, but the definition of this cosmological constant, the way of interpretation kept changing with time. Now the concept of this cosmological constant has been brought back to cosmology for many reasons. One of the reason is that, we have observations for energy density of different components of the universe (baryonic, dark matter, radiation), so we know that what this parameter is. Independent observations using cosmic microwave background shows that k=0. $$CMB, k=0: rho = rho_c = frac{3H_0^2}{8pi G} approx 10 : Hydrogen : atoms.m^{-3}$$ For k to be 0, $rho$ should be equal to $rho_c$, but everything we know if we add it up that does not give 0, which means that there is some other component that shows that it is much less than $rho_c$. $$rho = rho_b + rho_{DM} + rho_{rad} << rho_c$$ One more evidence of dark energy comes from the Type 1 Supernova Observation which occurs when white dwarf accretes the matter and exceeds the Chandrashekhar limit, which is a very precise limit (≈ 1.4M). Now every time when Type 1 Supernova Explosion occurs, we have the same mass which means that the total binding energy of the system is same and the amount of light energy we can see is the same. Of course, the supernova light increases and then faints, but if you measure the peak brightness it is always going to be the same which makes it a standard candidate. So, with a Type 1 Supernova we used to measure the cosmological component of the universe and astronomers found that the supernova with high red shift is 30% − 40% fainter than the low red shift supernova and it can be explained if there is any non-zero ∧ term. In cosmological models DE (Dark Energy) is treated as a fluid, which means that we can write the equation of state for it. The equation of state is the equation which connects the variables like Pressure, Density, Temperature, and Volume of two different states of the matter. Dimensionally we see, $$frac{8 pi G}{3}rho = frac{wedge}{3}$$ $$rho_wedge = frac{wedge}{8pi G}$$ Energy density of DE, $$epsilon_wedge = rho_wedge ast c^2 = frac{wedge c^2}{8 pi G}$$ Dark energy density parameter, $$Omega_wedge = frac{rho_wedge}{rho_c}$$ $Omega_wedge$ is the density of dark energy in terms of critical density. $$rho = rho_b + rho_{DM} +rho_wedge$$ There are a number of theories about dark energy, which is repelling the universe and causing the universe to expand. One hypothesis is that this dark energy could be a vacuum energy density. Suppose the space itself is processing some energy and when you count the amount of baryonic matter, dark matter and the radiation within the unit volume of space, you are also counting the amount of energy which is associated with the space, but it is not clear that the dark energy is really a vacuum energy density. We know that the relationship between density and scale factor for dark matter and radiation are, $$rho_m propto frac{1}{a^3}$$ $$rho_m propto frac{1}{a^4}$$ We have the density v/s scale factor plot. In the same plot, we can see that $rho_wedge$ is a constant with the expansion of the universe which does not depend on the scale factor. The following image shows the relationship between density and the scale factor. ‘ρ’ v/s ‘a’ (scale factor which is related to time) in the same graph, the dark energy is modelled as a constant. So, whatever dark energy we measure in the present universe, it is a constant. Points to Remember Independent observations using cosmic microwave background shows that k=0. $rho_wedge$ is a constant with the expansion of the universe which does not depend on the scale factor. Gravity is also changing

Cosmology – Fluid Equation In this chapter, we will discuss Fluid Equation and how it tells us regarding the density of universe that changes with time. Estimating ρc and ρ in the Present Universe For the present universe − $$rho_c simeq 10^{11}M_odot M_{pc}^{-3} simeq 10: hydrogen : atoms : m^{-3}$$ There is a whole range of critical density in our outer space. Like, for intergalactic medium $rho_c$ is 1 hydrogen atom $m^{-3}$, whereas for molecular clouds it is $10^6$ hydrogen atoms $m^{−3}$. We must measure $rho_c$ considering proper samples of space. Within our galaxy, the value of $rho_c$ is very high, but our galaxy is not a representative of the whole universe. So, we should go out to space where cosmological principle holds, i.e., distances ≈ 300 Mpc. Looking at 300 Mpc means looking 1 billion years back, but it is still the present universe. Surveys like SDSS are conducted to determine the actual matter density. They take a 5×500×5 Mpc3 volume, count the number of galaxies and add all the light coming from these galaxies. Under an assumption that 1 L ≡ 1 M, i.e. 1 solar Luminosity ≡ 1 solar Mass. We do a light to mass conversion and then we try to estimate the number of baryons based on the visible matter particles present in that volume. For example, $$1000L_odot ≡ 1000M_odot / m_p$$ Where, mp= mass of proton. Then we get roughly the baryon number density $Omega b ∼= 0.025$. This implies $rho b = 0.25%$ of $rho_c$. Different surveys have yielded a slightly different value. So, in the local universe, number density of visible matter is much less than critical density, meaning we are living in an open universe. Mass with a factor of 10 is not included in these surveys because these surveys account for electromagnetic radiation but not dark matter. Giving, $Omega_m = 0.3 − 0.4$. Still concludes that we are living in an open universe. Dark matter interacts with gravity. A lot of dark matter can halt the expansion. We haven’t yet formalized how $rho$ changes with time, for which we need another set of equations. Thermodynamics states that − $$dQ = dU + dW$$ For a system growing in terms of size, $dW = P dV$. Expansion of universe is modelled as adiabatic i.e. $dQ = 0$. So, volume change should happen from change in internal energy dU. Let us take a certain volume of universe of unit comoving radius i.e. $r_c = 1$. If $rho$ is the density of material within this volume of space, then, $$M = frac{4}{3} pi a^3r_c^3 rho$$ $$U = frac{4}{3}pi a^3rho c^2$$ Where, U is the Energy density. Let us find out the change in internal energy with time as the universe is expanding. $$frac{mathrm{d} U}{mathrm{d} t} = 4 pi a^2 rho c^2 frac{mathrm{d} a}{mathrm{d} t} + frac{4}{3}pi a^3 c^2frac{mathrm{d} rho}{mathrm{d} t}$$ Similarly, change in volume with time is given by, $$frac{mathrm{d} V}{mathrm{d} t} = 4pi a^2 frac{mathrm{d} a}{mathrm{d} t}$$ Substituting $dU = −P dV$. We get, $$4pi a^2(c^2 rho +P)dot{a}+frac{4}{3}pi a^3c^2dot{rho} = 0$$ $$dot{rho}+3frac{dot{a}}{a}left ( rho + frac{P}{c^2} right ) = 0$$ This is called the Fluid Equation. It tells us how the density of universe changes with time. Pressure drops as the universe expands. At every instant pressure is changing, but there is no pressure difference between two points in the volume considered, so, the pressure gradient is zero. Only relativistic materials impart pressure, matter is pressure-less. Friedmann Equation along with Fluid Equation models the universe. Points to Remember Dark matter interacts with gravity. A lot of dark matter can halt the expansion. Fluid Equation tells us how the density of universe changes with time. Friedmann Equation along with Fluid Equation models the universe. Only relativistic materials impart pressure, matter is pressure-less. Learning working make money

Cosmology – Robertson-Walker Metric In this chapter, we will understand in detail regarding the Robertson-Walker Metric. Model for Scale Factor Changing with Time Suppose a photon is emitted from a distant galaxy. The space is forward for photon in all directions. The expansion of the universe is in all the directions. Let us see how the scale factor changes with time in the following steps. Step 1 − For a static universe, the scale factor is 1, i.e. the value of comoving distance is the distance between the objects. Step 2 − The following image is the graph of the universe that is still expanding but at a diminishing rate, which means the graph will start in the past. The t = 0 indicates that the universe started from that point. Step 3 − The following image is the graph for the universe which is expanding at a faster rate. Step 4 − The following image is the graph for the universe that starts contracting from now. If the value of the scale factor becomes 0 during the contraction of universe, it implies the distance between the objects becomes 0, i.e. the proper distance becomes 0. The comoving distance which is the distance between the objects at a present universe, is a constant quantity. In the future, when the scale factor becomes 0, everything will come closer. The model depends on the component of the universe. The Metric for flat (Euclidean: there is no parameter for curvature) expanding universe is given as − $$ds^2 = a^2(t)left ( dr^2+r^2dtheta^2+r^2sin^2theta dvarphi^2 right )$$ For space–time, the line element that we obtained in the above equation is modified as − $$ds^2 = c^2dt^2 – left { a^2(t) left ( dr^2 + r^2dtheta ^2 + r^2sin^2theta dvarphi^2 right ) right }$$ For space – time, the time at which the photon is emitted and when it is detected is different. The proper distance is the instantaneous distance to objects which can change over time due to the expansion of the universe. It is the distance that the photon had travelled from different objects to get to us. It is related to the comoving distance as − $$d_p = a(t) times d_c$$ where $d_p$ is the proper distance and $d_c$ is the comoving distance, which is fixed. The distance measured to the objects in the present universe is taken as the comoving distance, which means the comoving distance is fixed and is unchanged by the expansion. For past, the scale factor was less than 1, which indicates that the proper distance was smaller. We can measure the redshift to a galaxy. Hence the proper distance $d_p$ corresponds to $c times t(z)$, where $t(z)$ is the lookback time towards a redshift and c is the speed of light in vacuum. The lookback time is a function of the redshift (z). Based on the above notion, let us analyze how cosmological red shift is interpreted in this scenario of $d_p = a(t) times d_c$. Assume a photon (which is earth bound) is emitted by the galaxy, G. The $t_{em}$ corresponds to the time when photon was emitted; $a(t_{em})$ was the scale factor at that time when the photon was emitted. By the time of detection of the photon, the whole universe had expanded, i.e. the photon is redshifted at the time of detection. The $t_{obs}$ corresponds to the time when the photon is detected and the corresponding scale factor is $a(t_{obs})$. The factor by which the universe has grown is given by − $$frac{a(t_{obs})}{a(t_{em})}$$ The factor by which wavelength has expanded is − $$frac{lambda_{obs}}{lambda_{em}}$$ which is equal to the factor by which the universe has grown. The symbols have their usual meaning. Therefore, $$frac{a(t_{obs})}{a(t_{em})} = frac{lambda_{obs}}{lambda_{em}}$$ We know that redshift (z) is − $$z=frac{lambda_{obs} – lambda_{em}}{lambda_{em}} = frac{lambda_{obs}}{lambda_{em}} – 1$$ $$1 + z = frac{a(t_{obs})}{a(t_{em})}$$ The present value of the scale factor is 1, hence $a(t_{obs}) = 1$ and denoting the scale factor when photon was emitted in the past by $a(t)$. Therefore, $$1 + z = frac{1}{a(t)}$$ Interpretation of Redshift in Cosmology To understand this, let us take the following example: If $z = 2$ then $a(t) = 1/3$. It implies that the universe has expanded by a factor of three since light left that object. The wavelength of the received radiation has expanded by a factor of three because the space has expanded by the same factor during its transit from the emitting object. It should be noted that at such large values of z, the redshift is mainly the cosmological redshift, and it is not a valid measure of the actual recessional velocity of the object with respect to us. For cosmic microwave background (CMB), z = 1089, which means that the present universe has expanded by a factor of ∼1090. The metric for flat, Euclidean, expanding universe is given as − $$ds^2 = a^2(t)(dr^2 + r^2dtheta^2 + r^2sin^2theta dvarphi^2)$$ We wish to write the metric in any curvature. Robertson and Walker proved for any curvature universe (which is homogeneous and isotropic), the metric is given as − $$ds^2 = a^2(t) left [ frac{dr^2}{1-kr^2} + r^2dtheta^2 + r^2sin^2theta dvarphi^2 right ]$$ This is generally known as the Robertson–Walker Metric and is true for any topology of space. Please note the extra factor in $dr^2$. Here 𝑘 is the curvature constant. Geometry of the Universe The Geometry of the Universe is explained with the help of the following Curvatures, which include − Positive Curvature Negative Curvature Zero Curvature Let us understand each of these in detail. Positive Curvature If a tangent plane drawn at any point on the surface of the curvature does not intersect at any point on the surface, it is called surface with a positive curvature i.e. the surface stays on one side of the tangent plane at that point. The surface of the sphere has positive curvature. Negative Curvature If a tangent plane drawn at a point on the surface of the curvature intersects at any point on the surface,

Redshift Vs. Kinematic Doppler Shift A galaxy which is at redshift z = 10, corresponds to v≈80% of c. The mass of the Milky Way is around 1011M&odot;, if we consider the dark matter, it is 1012M&odot;. Our milky way is thus massive. If it moves at 80% of c, it does not fit in the general concept of how objects move. We know, $$frac{v_r}{c} = frac{lambda_{obs} – lambda{rest}}{lambda_{rest}}$$ For small values of z, $$z = frac{v_r}{c} = frac{lambda_{obs}-lambda_{rest}}{lambda_{rest}}$$ In the following graph, the class between flux and wavelength, there are emission lines on top of the continuum. From the H-α line information, we get to conclude that roughly z = 7. This implies galaxy is moving at 70% of c. We are observing a shift and interpreting it as velocity. We should get rid of this notion and look at z in a different way. Imagine space as a 2D grid representing the universe as shown below. Consider the black star to be our own milky way and the blue star to be some other galaxy. When we record light from this galaxy, we see the spectrum and find out its redshift i.e., the galaxy is moving away. When the photon was emitted, it had relative velocity. What if the space was expanding? It is an instantaneous redshift of photon. Cumulative redshifts along the space between two galaxies will tend to a large redshift. The wavelength will change finally. It is the expansion of the space rather than the kinematic movement of the galaxies. The following image shows if mutual gravity overflows expansion then this is not participating in the Hubble’s law. In the Kinematic Doppler Shift, the redshift is induced in a photon at the time of emission. In a Cosmological Redshift, in every step, it is getting cumulatively redshifted. In a gravitational potential, a photon will get blue shifted. As it crawls out of gravitational potential, it gets redshifted. As per a Special Theory of Relativity, two objects passing by each other cannot have a relative velocity greater than the speed of light. The velocity we speak about is of the expansion of the universe. For large values of z, the redshift is cosmological and not a valid measure of the actual recessional velocity of the object with respect to us. The Cosmological Principle It stems from the Copernicus Notion of the universe. As per this notion, the universe is homogeneous and isotropic. There is no preferred direction and location in the universe. Homogeneity means no matter which part of the universe you reside in, you will see universe is the same in all parts. Isotropic nature means no matter which direction you look, you are going to see the same structure. A fitting example of homogeneity is a Paddy field. It looks homogeneous from all parts, but when wind flows, there are variations in its orientation, thus it is not isotropic. Consider a mountain on a flat land and an observer is standing on the mountain top. He will see the isotropic nature of the flat land, but it is not homogeneous. If in a homogeneous universe, it is isotropic at a point, it is isotropic everywhere. There have been large scale surveys to map the universe. Sloan Digital Sky Survey is one such survey, which did not focus much on the declination, but on the right ascension. The lookback time is around 2 billion years. Every pixel corresponds to the location of a galaxy and the color corresponds to the morphological structure. The green color represented blue spiral galaxy while the red false color indicated massive galaxies. Galaxies are there in a filamentary structure in a cosmological web and there are voids in between the galaxies. $delta M/M cong 1$ i.e., fluctuation of mass distribution is 1 M is the mass of the matter present within a given cube. In this case, take the volume 50 Mpc cube. For a cube side of 1000 Mpc, $delta M/M cong 10^{−4}$. One way to quantify homogeneity is to take mass fluctuations. Mass fluctuations will be higher at lower scales. For quantifying the isotropic nature, consider cosmic microwave background radiation. Universe is nearly isotropic at large angular scales. Points to Remember Two objects passing by each other cannot have a relative velocity greater than the speed of light. Cosmological Principle states that the universe is homogeneous and isotropic. This homogeneity exists at a very large angular scale and not on smaller scales. SDSS (Sloan Digital Sky Survey) is an effort to map the night sky, verifying the Cosmological Principle. Learning working make money

Redshift and Recessional Velocity Hubble’s observations made use of the fact that radial velocity is related to shifting of the Spectral Lines. Here, we will observe four cases and find a relationship between Recessional Velocity ($v_r$) and Red Shift (z). Case 1: Non-Relativistic Case of Source moving In this case, v is much less than c. Source is emitting some signal (sound, light, etc.), which is propagating as Wavefronts. The time interval between the sending of two consecutive signals in the source frame is Δts. The time interval between the reception of two consecutive signals in the observer frame is Δto. If both the observer and source are stationary, then Δts = Δto, but this is not the case here. Instead, the relation is as follows. $$Delta t_o = Delta t_s + frac{Delta l}{c}$$ Now, $Delta l = v Delta t_s$ Also, since (wave speed x time) = wavelength, we get $$frac{Delta t_o}{Delta t_s} = frac{lambda_o}{lambda_s}$$ From the above equations, we get the following relation − $$frac{lambda_o}{lambda_s} = 1 + frac{v}{c}$$ where $lambda _s$ is the wavelength of the signal at the source and $lambda _o$ is the wavelength of the signal as interpreted by the observer. Here, since the source is moving away from the observer, v is positive. Red shift − $$z = frac{lambda_o – lambda_s}{lambda_s} = frac{lambda_o}{lambda_s} – 1$$ From the above equations, we get Red shift as follows. $$z = frac{v}{c}$$ Case 2: Non-Relativistic Case of Observer Moving In this case, v is much less than c. Here, $Delta l$ is different. $$Delta l = v Delta t_o$$ On simplification, we get − $$frac{Delta t_o}{Delta t_s} = left ( 1 – frac{v}{c} right )^{-1}$$ We get Red shift as follows − $$z = frac{v/c}{1-v/c}$$ Since v << c, the red shift expression for both Case I and Case II are approximately the same. Let us see how the red shifts obtained in the above two cases differ. $$z_{II} – z_I = frac{v}{c} left [ frac{1}{1 – v/c}-1 right ]$$ Hence, $z_{II} − z_{I}$ is a very small number due to the $(v/c)^2$ factor. This implies that, if v << c, we cannot tell whether the source is moving, or the observer is moving. Let us now understand the Basics of STR (Special Theory of Relativity) − Speed of light is a constant. When the source (or observer) is moving with a speed comparable to the speed of light, relativistic effects are observed. Time dilation: $Delta t_o = gamma Delta t_s$ Length contraction: $Delta l_o = Delta t_s/gamma$ Here, $gamma$ is the Lorrentz factor, greater than 1. $$gamma = frac{1}{sqrt{1-(v^2/c^2)}}$$ Case 3: Relativistic Case of Source Moving In this case, v is comparable to c. Refer to the same figure as in Case I. Due to the relativistic effect, time dilation is observed and hence the following relation is obtained. (Source is moving with relativistic speed) $$Delta t_o = gamma Delta t_s + frac{Delta l}{c}$$ $$Delta l = frac{vgamma Delta t_s}{c}$$ $$frac{Delta t_o}{Delta t_s} = frac{1 + v/c}{sqrt{1- (v^2/c^2)}}$$ On further simplification, we get, $$1 + z = sqrt{frac{1+v/c}{1-v/c}}$$ The above expression is known as the Kinematic Doppler Shift Expression. Case 4: Relativistic Case of Observer Moving Refer to the same figure as in Case II. Due to relativistic effect, time shortening is observed and hence the following relation is obtained. (Observer is moving with relativistic speed) $$Delta t_o = frac{Delta t_s}{gamma}+frac{Delta l}{c}$$ $$Delta l = frac{vDelta t_o}{c}$$ $$frac{Delta t_o}{Delta t_s} = frac{sqrt{1-( v^2/c^2)}}{1-v/c}$$ On further simplification, we get − $$1 + z = sqrt{frac{1+ v/c}{1- v/c}}$$ The above expression is the same as what we got for Case III. Points to Remember Recessional velocity and redshift of a star are related quantities. In a non-relativistic case, we cannot determine whether the source is moving or stationary. In a relativistic case, there is no difference in the redshift-recessional velocity relationship for source or observer moving. Moving clocks move slower, is a direct result of relativity theory. Learning working make money

Horizon Length at the Surface of Last Scattering Horizon length is the distance travelled by light photons from ‘The Big Bang’ to ‘The Recombination Era’. The 1st peak of the angular spectrum is at θ = 1◦ (l = 180), which is a very special length scale. The proper distance between two points is given by − $$r_p = int_{0}^{t}cdt$$ When we take the time frame of t = 0 to t = trec, then $$r_H = int_{0}^{t_{rec}}cdt$$ Where $r_H$ is the proper horizon distance. Now, we know that − $$dot{a} = frac{mathrm{d} a}{mathrm{d} t}$$ $$dt = frac{da}{dot{a}}$$ When t = 0, a = 0. Then $t = t_{rec}, a = a_0 / (1 + z_{rec})$. Hence, we can write, $$r_H(z_{rec})=int_{0}^{a_{rec}} cfrac{da}{aH}$$ $$H(a_{rec}) = H(z_{rec}) = H_0sqrt{Omega_{m,0}}a^{-3/2}$$ During the Recombination period universe was matter dominated. i.e., Ωrad << Ωmatter. Therefore, the term radiation is dropped. $$r_H(z_{rec}) = frac{c}{H_0sqrt{Omega_{m,0}}}int_{0}^{a_{rec}} frac{da}{a^{-1/2}}$$ $$r_H(z_{rec}) = frac{2c}{3H_0sqrt{Omega_{m,0}}}frac{1}{(1+z_{rec})^{3/2}}$$ $$theta_H(rec) = frac{r_H(z_{rec})}{d_A(z_{rec})}$$ Which is equal to 0.5 degrees, if we put all the known values in the equation. The Electromagnetic radiation is opaque from the surface of last scattering. Any two points ‘not’ lying within the horizon of each other need not have the same properties. So, it will give different temperature values. We can get two points on this surface which did not intersect with each other, which means at one point the universe expanded faster than the speed of light which is the inflationary model for expansion. Points to Remember The horizon length is the distance travelled by light photons from ‘The Big Bang’ to ‘The Recombination Era’. During the Recombination period, the universe was matter dominated. Electromagnetic radiation is opaque from the surface of last scattering. Learning working make money

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