Basic notions of Cosmology

Cosmology studies the origin and evolution of the Universe. It is beyond the scope of this book to give a full review of Cosmology, but we will see its impact in, for example, calculating distances.

Introduction

Cosmology is a very fascinating field that tries to deal with very profound questions, such as "What is the Universe made of?" ; "Was there a beginning?"; "Is the Universe static or is it changing? and if so, what is the future of the Universe?". In summary, questions that might keep you awake at night and your head spinning!

The field begun with Albert Einstein's theory of General Relativity, which provided the mathematical framework for the description of the Universe as a whole. After about a century, it is now widely accepted that ordinary atoms make up a tiny fraction of the energy density of the Universe and that two mysterious components, dark matter and dark energy have a nature requiring further understanding. Dark matter adds gravity to the Universe on the scale of galaxies and clusters of galaxies and dark energy acts on the larger scales and causes the accelerated expansion of the Universe. The current model that better describes our Universe is known as the $\rm \Lambda CDM$ model, where $\Lambda$ indicates the dark energy and CDM stands for cold dark matter. This is sometimes referred as the Standard Model of Cosmology . We draw it in big lines.

The Expansion of the Universe

While stars look motionless, they move at speeds between $1-10^2$ km/s. This means that they can travel to up $10^{10}$ km in 1 year but the closest star is at only about $10^{13}$ km, so at this distance it appears steady. Barnard's star is at $56 \times 10^{12}$ km, moves across the line of sight at 89 km/s and its ‘proper motion’ in 1 year is of 0.0029°. As long as we measure galaxies within about a hundred million light-years of our own, we find that the universe will not have changed much in the time it took light to travel from those galaxies to us.

But what if you're measuring a galaxy that's a few billion light-years away? In that case, the universe has changed significantly as the light has traveled.

Today, almost about a century after Hubble's discovery, three exquisite observations form the foundations of modern cosmology: the observation of the cosmic microwave background, the modern version of Hubble's diagram, and the measurement of the light elements synthetized in the first few minutes of the universe. We now have confidence that a geometrically flat universe has been expanding for the past $\sim 14 \rm Gyr$ , growing in contrast to the action of gravity from a hot and smooth Big Bang to the universe of galaxies, stars, planets we see. Observations have forced us to accept a dark and exotic universe that is composed by about ~27% of dark matter and only 5% of ordinary matter made up of protons and neutrons. The remaining ~68% is the mysterious dark energy, an unknown force which accelerates the expanding Universe. The plot below shows the composition of the Universe as measure by WMAP (“before Planck”) and as measured by Planck (“after Planck”).

Credit: ESA & the Planck Collaboration

In 1929 Edwin Hubble (see his original paper) observed spectral lines from distant galaxies with the 100-inch Mount Wilson telescope. Star spectra exhibit absorption lines at well defined frequencies depending on atoms composing them, depending on the energies required to excite electrons in the atomic energy levels. Spectral lines were shifted towards the red end of the spectrum and the amount of shift depended on the apparent brightness and hence on their distance.compared. An object in motion with respect to the observer has its radiation shifted in wavelength according to the following formula (Doppler shift - see Special Relativity and High-Energy Astrophysical Phenomena):

$$\lambda_{obs} = \lambda_{source} \sqrt{\frac{1+\beta}{1-\beta}} = \lambda_{source} \gamma (1+\beta)$$

Where to obtain the last equation we used the definition of Lorenz factor and multiplied the denominator and numerator under square root by ( $1+\beta$ ).The redshift is defined as $z = \frac{\Delta \lambda}{\lambda{source}} = \frac{\lambda_{obs}-\lambda_{source}} {\lambda_{source}}$ .

Then $(1+z) = \gamma (1+\beta)$. This formula should be used for distant galaxies. For z < 0.003 we can consider for slow expansion: $\beta <<1$ and so $\gamma \sim 1 \Rightarrow z \sim \beta = \frac{\rm v}{c}.$

Interpreting the observations as Doppler shift, Hubble discovered that galaxies recede from us in all directions and that more distant ones (for which local motion, resulting from gravitational bound states of galaxies, is negligible) recede more rapidly proportionally to their distance, according to the linear law:

$v = H_0 D_L$.

This is the Hubble's Law, and the constant was named (you guessed it!) the Hubble Constant. We will see that this constant is not really a constant and its value is still controversial.

Regardless of the implications to the state of the Universe, the Hubble's Law makes it possible to estimate distances from the redshift. In particular we have that:

$$D_L \simeq \frac{c}{H_0}z \simeq 4000 {\rm\; Mpc}\cdot z$$

So Hubble's Law can be considered the final step of the Cosmic Ladder. The original Hubble graph was done with galaxies out to $2 {\rm \;Mpc}$. Current experiments like the Hubble Key Project have tested the correlation up to $23 {\rm \;Mpc}$.

Note that $H_0$ has units of time$^{-1}$however, we explicitly write the other dimensions to better understand its meaning.

The plot below (from his paper and reanalized in the 2004 review by R.P. Kirshner and the 2015 N. A. Bahcall review) showed the radial velocity corrected for solar motion - distance (notice the wrong units) among extra-galactic nebulae. Distances were estimated from apparent luminosities F at Earth, knowing the intrinsic luminosity L: $F = \frac{L}{4\pi D_L^2}$ (see also Introduction to astronomical quantities). In the plot, the black discs and full line represent the solution for solar motion by using the nebulae individually; the circles and broken line represent the solution combining the nebulae into groups; the cross represents the mean velocity corresponding to the mean distance of 22 nebulae whose distances could not be estimated individually.

The plot below is known as Hubble's diagram: the slope of the line at low redshifts determines the present normalized expansion rate, i.e. the Hubble constant. The shape of the trend at large redshiftsdetermines the global geometry of the universe.

Hubble's graph of redshift versus distance.

As you can see there are two remarkable results: one is that the spectral lines of galaxies were all redshifted, meaning that most galaxies were moving away from us. The second unexpected result is that that the farther away galaxies are, the faster they seem to recede from us. Of course this could not be explained with the formula of the relativistic Doppler Effect. The accepted explanation was that the Universe was expanding, but not in the sense of things moving away from each other. What is happening is that space-time is being created in between galaxies, so they appear to be moving away from us. This source of redshift is called the Cosmological Redshift or Hubble Flow.Sources at cosmological distances (where local motion, resulting from gravitational bound states of galaxies, are negligible) move away at speeds proportional to their distance. Close-by galaxies may look approaching due to the relative motion (e.g. for Andromeda it looks we are approaching it but it is at only 2.5 Mly). Globally the universe is in a state of expansion, following a violent explosion and galaxies are rushing apart at speeds approaching the speed of light and even surpassing it. As a comparison, introduced by Martin Gardner, a popular science writer who was also a longtime columnist for Scientific American, is to imagine a gigantic blob of dough (representing space) with a bunch of raisins embedded throughout (representing galaxies). Now, if someone puts the dough in the oven, it will expand or, more accurately, stretch, keeping the same proportions as it had before, but with all the distances between raisins getting bigger as time goes on.

The Hubble constant value (see plot below from here) is given by the recent Planck 2015 and Planck 2018 results: $H_0 = 67.4 \pm 0.66$km s$^{-1}$ Mpc$^{-1}$, with an error of about 1% (you find a useful table of cosmology constants here). The Hubble constant in this case is estimated from the cosmological model that fits observations of the cosmic microwave background (CMB), which come from the very young Universe. In the figure below the units give the velocity of the expansion in km/s for every million parsecs (Mpc) of separation in space, where a parsec is equivalent to 3.26 light-years. The Hubble Stace Telescope (HST) obtained in 2001:$H_0 = 72 \pm 8$km s$^{-1}$ Mpc$^{-1}$. This study used the empirical period–luminosity relation for Cepheid variable stars, and also calibrated a number of secondary distance indicators (Type Ia Supernovae(SNe Ia), the Tully–Fisher relation, surface-brightness fluctuations, and Type II Supernovae). There is some tension between the $H_0$ values from Planck and the traditional cosmic distance ladder methods. For example, the SH0ES' results deviate from Planck by 4.4σ. SHOES extended the 2001 results of HST based on 70 long-period Cepheids in the Large Magellanic Cloud (LMC), combined with Milky Way parallaxes and masers in NGC4258, yielding $H_0 = 74.0 \pm 1.4 \rm km s^{-1} Mpc^{-1}$. The major sources of uncertainty in this result are thought to be due to the heavy element abundance of the Cepheids and the LMC distance relative to which the distance Cepheid distance is measired. The possible trend for higher $H_0$ values from the nearby Universe and a lower $H_0$ from the early Universe has led some researchers to propose a time-variation of the dark energy component or other exotic scenarios.

Collection of measurements of the Hubble Constant (Copyright: ESA/Planck Collaboration)

Another recent collection of values is preseted here:

Very recently there the Hubble constant has beed measured also using gravitational waves to be $68^{+14}_{-7}$ km s$^{-1}$ Mpc$^{-1}$, namely a speed for the galaxy NGC4993 of $v_{H} = cz = 3017 \pm 166$ km/s. The result is in the image below from B.P. Abbott et al., arXiv:1908.06060. An electromagnetic counterpart to the gravitational wave signal from the coalescence of 2 neutron starts, GW170817, has been observed, leading to the first standard-siren measurement of $H_0$. The figure below from this paper shows the results from the black hole mergers (red), for which a number of galaxies indentification was possible, compared to the result of GW170817, univocally localized due to an independent electromagnetic measurement of the distance. It is seen that the Joint result is mostly influenced by GW170817 and te result is compared to Planck and supernovae observations, SH0ES.

Measurements of Hubble constant from gravitational wave compared to CMB abd optical results.

We observe that, if you were to measure the "distance" to a galaxy at cosmological distance, is the distance that you measure the distance the light was emitted? Or the distance the light traveled to reach us (which includes some extra distance due to the universe expansion while the light was moving through it)? Or would you measure the distance that the galaxy is currently from us, which is the largest of them all? The Hubble constant changes with time, and depending on how it changes, individual galaxies might be speeding up or slowing down. Redshift is what astronomers typically measure: as light travels through the expanding universe, the light gets stretched by the same factor that the universe does, causing its wavelength to increase.

Notice that, as written above, Hubble's law predicts superluminal recession at distances $D_L > c/H_0$(for a detailed discussion see arxiv:0310808). Hubble expansion does not contraddict special nor general relativity of course! There is no contradiction with special relativity if superluminal motion occurs outside the observer’s inertial frame (general relativity was specifically derived to be able to predict motion in non-inertial frames). Galaxies that are receding from us superluminally are at rest locally (their peculiar velocity $v_{pec} = 0$) and motion in their local inertial frames is well described by special relativity. Simply, the galaxies and the photons are both receding from us at recession velocities greater than the speed of light. From the paper above the following diagram shows the velocity as a function of redshift. The low redshift approximation is the one calculated above: $v = cz$

The Age and Center of the Universe

The center of the Universe is everywhere.

Giordano Bruno

By observing all galaxies receding from us, one is tempted to think that we are in a short of special location in the Universe, but we are not. The common picture is that galaxies are like dots in the surface of a ballon that is being inflating. From every dot in your balloon-surface Universe you will see all dots receding from you, misleading you to think that you are at the centre of the Universe.

The Cosmological Principle

The cosmological principle, formulated by Einstein, is the notion that the distribution of matter in the Universe is homogeneous and isotropic when viewed on a large enough scale. It basically states that all spatial positions in the Universe are essentially equivalent, despite when we look out of the window at night the Universe looks anything but uniform and isotropic: we see group stars, the Milky Way, and not the same structures in all directions. Despite the matter today does show enormous fluctuations in density in the form of stars, galaxies and larger structures, the average separation between galaxies is of order 100 times their diameter and the overall expansion of the universe of billions of galaxies on large enough scales (>> of intergalactic separations) still appears to be reasonably well described by the FLRW model. Hence, statistically, at large scale, the Universe actually looks rather uniform and isotropic, such as the cosmological principle states.

What does it mean homogeneous and isotropic? Although these two things might look similar they are not, homogeneity means that the distribution of matter that an observer can see in any point of the universe is the same and it is the same even in each epoch. Isotropy means looking in any direction the universe seems the same. The principles are distinct but closely related, because a universe that appears isotropic from any two (for a spherical geometry, three) locations must also be homogeneous. But in general one can have an isotropic Universe that is not homogenous. The best evidence for isotropy and homogeneity on all scales in fact comes from observations of the cosmic microwave background (CMB), a photograph ' of the distribution of matter and radiation as it appeared in the universe ~380'000 years old.

The Friedmann–Lemaître-Robertson-Walker model of the universe (FLRW for short) or `Standard Model' of present cosmology is assumes a competely isotropic and homogeneous distribution of matter and radiation, behaving as a frictionless fluid, in expansion with a uniform space–time curvature. The FLRW line element (or FLRW space-time metric) describes the overall geometry and evolution of the Universe in terms of 2 cosmological parameters accounting for the spatial curvature and the overall expansion (contraction). It is given by (taking c = 1):

$$ds^2 = dt^2 - R^2(t)\left[\frac{dr^2}{1-kr^2}+ r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2\right] \, \, (*)$$

where $R(t)$ is an expansion parameter or cosmological scale factor, that we assume here with units of distance. An alternative is to define a dimensionless scale factor $a(t) = \frac{R(t)}{R_0}$ with dimenson of and the parameter k describes the curvature of space. k = +1 corresponds to closed universe, k = −1 to open universe and k = 0 to the flat Euclidean space of special relativity.

Distances between objects (or points on the ballon) can be expressed as:

$$D(t) = R(t)r$$

where $R(t)$ multiplies the radial space coordinate r = comoving coordinate distance has no time dependence and is defined in a reference frame co-moving frame (i.e. enlarging ) with the expansion. Comparing the actual value R(0) to the value at time t: $R(t) = \frac{R(0)}{1+z}$, the reciprocal of the redshift factor. Sometimes its better to explicitly use a normalized scale factor as $a(t) = R(t)/R(0)$so that a(t) is a parameter quantifying the expansion. If we assume that the rate of expansion (ie. the Hubble constant) is essentially constant (it is not) the age of the Universe can be estimated by integrating both sides of the Hubble law:

$$\frac{{\rm d}R}{{\rm d}t} = H_0 R \rightarrow \int \frac{1}{R}{\rm d}R =\int H_0 {\rm d}t$$

$$\log R = H_0 t \rightarrow D(t) =const \cdot e^{H_0 t}$$

The distance between galaxies $D$ increase exponentially with time.

In this constant expansion rate, the Universe increases by a factor $e$ every $t_{\mathrm{Hubble}} = \frac{1}{H_0} = 14\times 10^9\rm{yr}$ which is known as the the Hubble time.

Notice that Hubble expansion is relevant only for cosmological distancies due to the smallness of $H_0$ . The Hubble also implies an acceleration of the Universe: $\rm v = H D \rightarrow g_{Hubble} = H^2 D = 5 \times 10^{-36} D[m] s^{-2}$ , where the distance D is measured in meters.

Notice that for distances as small as the distance between the Sun and the Earth: $D_{Sun-Earth} \sim 150 \times 10^9 \, \rm m \Rightarrow g_{Hubble} \sim 7 \times 10^{-25} \, \rm m/s^2$

We compare this number to the gravitational acceleration of Earth in solar orbit: $g_{Earth-Sun} = \frac{G M_\oplus}{D_{Sun-Earth}^2} = \frac{6.67 \times 10^{-11} \times 2 \times 10^{30}}{(150 \times 10^9)^2} = 6 \times 10^{-3} \, \rm m/s^2 \sim 10^{22} g_{Hubble}$ calculated at the same distance scale.

Only for a galactic mass of $10^{11}M_\odot$ and Mpc distances we find that the inward gravitational acceleration is $g_{galactic} = \frac{G \cdot 10^{11}M_{\odot}}{D_{Mpc}^2} = \frac{6.67 \times 2 \times 10^{30}}{(3 \times 10^{22})^2} \sim 10^{-14}$ is smaller than the outward Hubble acceleration $g_{Hubble} \sim 10^{-13}.$

Einstein Equations

It was Friedmann and then Lemaître who realized that making an isotropic, and uniform Universe provides a solution to Einstein's equation. Without the ambition to derive them, the cosmological equations of motion are derived from Einstein's equations of General Relativity:

$$\mathcal{R}_{\mu\nu} - \frac{1}{2}g_{\mu\nu} \mathcal{R} - \Lambda g_{\mu\nu}= 8 \pi G \mathcal{T}_{\mu\nu}$$

There is a reason why Einstein fields equations are usually written this way. The parameter $\mathcal{T}_{\nu\mu}$ is called the energy-momentum (in Newtonian mechanics mass is the source of gravity, in general relativity is something a bit more complex called the energy-momentum tensor), $\mathcal{R}_{\mu\nu}$ is the Ricci's tensor, $g_{\mu\nu}$ is the metric of space time described in equation (*). Dimensions are $\rm length^{-2}$. The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter–energy content of spacetime. It was Einstein that realized that his equations could be applied to a the Universe as a whole, for that one has to assume a given energy-momentum tensor, and the easiest one was to think about energy content of the Universe as an uniform, isotropic, fluid.

Friedmann equations

It is common to assume that the matter content of the Universe is a perfect fluid, for which

$$\mathcal{T}_{\mu\nu} = (\rho + p)U_\mu U_\nu - pg_{\mu\nu}$$

where $U = (1,0,0,0)$ is the fluid four-velocity vector in co-moving coordinates and $\rho$ is an energy density, namely the total density of matter, radiation and vacuum energy. The FLRW metric solution to the Einstein equations (when the FLRW metric is inserted in the Einstein's equations) can be reduced to the two Friedmann equations:

$$H^2(t) \equiv \left(\frac{\dot R}{R}\right)^{2} = \frac{8\pi G}{3}\rho - \frac{k}{R^2} + \frac{\Lambda}{3}$$

where $\Lambda$ is the cosmological constant and the last term is called the curvature term. Notice that General Relativity foresees that, in the presence of gravitating masses, the flat Euclidean space of special relativity is curved space/time. The path of maximum proper time are called geodesics. Geodesics are straight linesin Euclidean space but in the presence of masses paths are curved (also for photons). The other equation is:

$$\frac{\ddot R}{R} = \frac{\Lambda}{3}-\frac{4\pi G}{3}(\rho + 3p)$$

From these two equation (both with dimension $time^{-2}$ ) one can also obtain $\dot{\rho} = -3H (\rho + p)$ (as shown below) that is also obtainable from energy-conservation and from the first law of thermodynamics.

Neglecting the cosmological constant term ( $\Lambda = 0$ ), in a classical mechanics analogy treated in detail below, we can interpret $-\frac{k}{R^2}$ as total energy, and the evolution of the universe is governed by a competition of the potential energy $\frac{8\pi G}{3}\rho$ and the kinetic term $\frac{\ddot{R}}{R}$. Hence, the Universe is expanding or contracting and its ultimate fate is for$k = +1$ a re-collapse in a finite time, while for $k = 0, -1$ it will indefinitely expand. For $\Lambda \ne 0$ , these conclusions maybe altered.

When can define $\Lambda = 0$, we can define a critical density such that $k = 0$:$\rho_c = \frac{3H^2}{8\pi G} = 1.88 \times 10^{-26} h^2 {\rm kg \, m^{-3}}.$For the value of $H_0$ measured by Planck reported above $\rho_{c} = 8.5 × 10^{−27} \rm \, kg/m^3$and then $\rho_c c^2 \sim \, 4.8 \rm GeV m^{-3} .$

We define also the cosmological density parameter or closure parameter (the energy density relative to the critical energy): $\Omega_{tot} = \frac{\rho}{\rho_c},$ which is >1 for $k = + 1$ (closed Universe), $\Omega_{tot} = 1$ for the flat universe (current preferred value), and <1 for the open Universe. Present masurements lean towards a flat Universe.

With this definition, one can write the Friedmann equation at present time as $\frac{k}{R^2} = \frac{8\pi G}{3}\rho -H_0^2 \Rightarrow H_0^2 \frac{8\pi G}{3H_0^2}\rho = H_0^2 + H_0^2\frac{k}{R^2(0) H_0^2}$

$$\Rightarrow \Omega_{tot} = \frac{\rho}{\rho_c} = 1 + \frac{k}{R^2(0) H_0^2}$$

with the total value of the closure parameter determined by matter, radiation and dark and we consider in $\Omega_{tot}$ also the cosmological constant term as $\Omega_\Lambda = \frac{\Lambda}{3H_0^2}$.

Classical derivation of the Friedmann equation

Let's consider the case when the energy density of the Universe is dominated by non-relativistic matter: we consider a point of mass m accelerated by gravity at the surface of a sphere of radius D, density $\rho$ and mass $M = \frac{4}{3}\pi D^3 \rho$. We assume a spherically symmetric and homogeneous distribution of matter outside of the sphere. It will not contribute to the force in this case and the gravitational field at the surface of the sphere will be as if all the mass is in its centre. The equation of motion is:$m\ddot{D} = - G \frac{mM}{D^2}$ (**).

Since $D(t) = r R(t) \Rightarrow m r \ddot{R} = - G \frac{mM}{r^2 R^2}$ , all constant factors $r$ cancel out since $M\propto r^3$ and take for brevity $r = 1$ we get after multiplication by $\dot{R}$ on both sides and then time integration $\Rightarrow \frac{m\dot{R}^2}{2} - G\frac{mM}{R} = {\rm const} = -\frac{mk}{2}$ , where the term on the right represent in fact the total energy. You can verify this equation since $ \frac{{\rm d} (\dot{R}^2/2) }{{\rm dt}} = \dot{R}\ddot{R}$ and $\frac{{\rm d}R^{-1}}{{\rm d} t} = -R^{-2} \dot{R}$ .

The energy density

The total density is composed by 3 terms: $\rho = \rho_m + \rho_r + \rho_\Lambda$. To understand its contributions we can derive the second of the Friedmann's equations. The conservation of energy in a volume element dV for the cosmic fluid can be expressed as: dE = -p dV, with p = pressure. We forget for a monent that we used c = 1 everywhere to understand all elements. The energy density is also $E/V = \rho c^2 \Rightarrow d(\rho c^2 R^3) = -p {\rm d}(R^3) \Rightarrow \dot{\rho}c^2 R^3 + 3 \rho c^2 R^2 \dot{R}= - 3p R^2 \dot{R}$ (having differentiated). Hence $\dot{\rho} = - 3\frac{\dot{R}}{Rc^2}(\rho c^2 + p)$. With c = 1 we have $\dot{\rho} = - 3\frac{\dot{R}}{R}(\rho + p)= -3 H^2 (\rho + p)$ (++).

If we differentiate the first Friedmann's equation for $k = 0$ :

$\left(\frac{\dot R}{R}\right)^{2} = \frac{8\pi G}{3}\rho + \frac{\Lambda}{3} (***) \Rightarrow 2 \dot{R}\ddot{R} = \frac{8\pi G \dot{\rho} R^2}{3} + \frac{16\pi G \rho R \dot{R}}{3}$

We substitute the expression above of $\dot{\rho}$ and obtain:

$$2 \dot{R}\ddot{R} = - 8\pi G \dot{R}R (\rho+p) + \frac{16\pi G \rho R \dot{R}}{3} \Rightarrow \ddot{R} = - \left(\frac{8 \pi GR}{3}\right) (\rho +3p)$$

This would be the same as equation (**) for p = 0 with $\rho = \frac{M}{\frac{4}{3}\pi R^3}$for non-relativitic matter (see above).

In general $\rho$ and p are connected by an equation of state parameter: $w = p / \rho$ which can be constant for matter, radiation and the vacuum state or maybe time dependent. If it is time indepenent one obtains: $\dot{\rho} = -3 \frac{\dot{R}}{R}(1+w)\rho \Rightarrow \rho \propto R^{-3(1+w)}.$

For the matter dominated universe we already demonstrated above that $R \propto t^{2/3}$ and so being $w \sim 0$ (below we calculate it precisely) we obtain $\rho \propto R^{-3} \propto t^{-2}$ .

In the hot and dense early universe, it is appropriate to assume a gas of radiation with $w = 1/3$and so $\rho \propto R^{-4}$. If we neglect at early times (R small) the curvature term $k/R^2$ and the cosmological constant, in the radiation dominated universe we obtain the Friedmann equation $H^2(t) \equiv \left(\frac{\dot R}{R}\right)^{2} = \frac{8\pi G}{3}\rho \Rightarrow \dot{R}^2 = \frac{8\pi G}{3R^2}$ so $R{\rm d}R = \sqrt{\frac{8\pi G}{3}} dt\Rightarrow R \propto t^{½}.$

In summary,

Dominant Regime	Equation of state	Energy density	Expansion parameter
Radiation	$p = \frac{\rho}{3}$	$\rho \propto R^{-4} \propto t^{-2}$	$R \propto t^{1/2}$
Matter	$p = \left(\frac{2}{3}\right)\rho \times \rm v^2$	$\rho \propto R^{-3} \propto t^{-2}$	$R\propto t^{2/3}$
Vacuum	$p = -\rho$	$\rho = const$	$R(t) \propto \exp({\alpha}t)$

To demonstrate the values used above for matter and radiation dominated universe one can also use an ideal gas of particles with mass m and velocity v, confined in a cubical box of side L, where particles collide on its walls. A particle will with component of speed along the x axis $\rm v_x$ will strike one wall at a rate $\frac{1}{\Delta t} = \rm v_x/2L$ and the rate of change of momentum will be $\Delta P_x = P_x - (-P_x) = 2m \rm v_x$, Hence the force exerted by the particle will be $F_x = \frac{\Delta P_x}{\Delta t} = \rm \frac{m v_x^2}{L}$ and then the pressure on a face of area $A = L^2$is $p = \frac{m\rm v_x^2}{L^3}$ and the total pressure on the face A of the cube from n = number of particles per unit volume $= N/L^3$ is $p = nm< \rm v_x^2>$and because the gas is isotropic $\rm <v_x> = <v_y> = <v_z>$ , then $<\rm v^2> = 3<\rm v_x^2>$ . So we obtain a pressure of $p = \frac{1}{3} m n \rm <v^2> = \frac{1}{3}n <Pv>$.

For non relativistic particles: the kinetic energy density is $\epsilon = \frac{1}{2} mn <v^2>.$ Hence $p_{non \, rel.} = \frac{2}{3}\rho c^2 \times \frac{\rm v^2}{c^2}.$ Since $\rm v << c$ , then, as anticipated above, that $p \sim 0$ for non relativistic matter. For radiation we have: $\rm v = c$ and so $\rho c^2 = mn c^2 = n <Pc> \Rightarrow p_{rel} = \frac{\rho c^2}{3}.$ Taking again c = 1 we understand the factor $w = \frac{1}{3}.$

Strangely as it may seem from a classical physics point of view where vacuum contains nothing, here the vacuum may contain an energy density and exert a pressure equivalent to a gravitational repulsion. If in the Friedmann's equation above (***) $\Lambda$ is large at large R this term will dominate and the expansion will become exponential : $R(t) \propto \exp({\alpha}t)$ with $\alpha = (\frac{\Lambda}{3})^{1/2}$ . The relations for the vacuum in the table above can be demonstrated if we assume a piston enclosing an isolated cylinder filled with vacuum state energy of energy density $\rho c^2$. If the piston is moved out adiabatically by the element of volume dV, an extra-vacuum energy is created: $\rho c^2 dV$ and the work supplid by the vacuum pressure $pdV$. This must be supplied by the work done by the vacuum pressure, PdV. Hence, for energy conservation $p = -\rho c^2$ (meaning $w = -1$ and the measured value by Planck is $w_0$ = −1.03 ± 0.03, consistent with a cosmological constant ) and due to equation (++) above ${\dot{\rho}} =0 \Rightarrow \rho = const$and a negativ e pressure corresponds to a repulsion.

This was first observed from very distant supernove Type Ia in 1998 by two groups, that realised that distant SN Ia shine less strongly than expected (Perlmutter S et al 1999 Astrophys. J. 517 565 - Supernova Cosmology Project, Schmidt B et al 1998 Astrophys. J. 507 46 - High-Z Supernova Search). This was explained or by the fact that these SNs, exploded billions of years ago, had traveled greater distances than expected, indicating that the Universe is speeding up. SN I and SN II are so classified depending on whether they show hydrogen lines in their spectra at maximum light or not. The hydrogen-free type Ia supernovae are attributed to the thermonuclear detonation of white dwarf stars (so very ancient ones!) and the type II (as well as SN Ib and Ic) to the core collapse of massive stars. The SN Ia are thought to leave no stellar remnant while the SN II, SN Ib and Ic are responsible for the formation of neutron stars and stellar-mass black holes. Despite their very different origins and mechanisms, the intrinsic luminosity of both types is comparable and their combined rate is ~a few per century in a galaxy like ours. An example of SN Ia is probably the Tycho’s supernova of 1572 in the Milky Way, while SN 1987A in the Large Magellanic Cloud was a variant of the SN II class. This effect is not due by aborption of light by powders as initially advocated. In the past few years, astronomers have solidified the case for cosmic acceleration by studying ever more remote supernovae.

The modern Hubble diagram then looks like the plot from Perlmutter et al, 1999, which shows the Hubble diagram for 42 high-redshift type Ia supernovae from the Supernova Cosmology Project and 18 low-redshift type Ia supernovae from the Calàn/Tololo Supernova Survey, plotted on a linear redshift scale to display details at high redshift. The second plot from the bottom shows the residuals from the best fit flat cosmology model with $(\Omega_m,\Omega_\Lambda) = (0.28, 0.72)$ . The bottom plot the uncertainty-normalized residuals.

An even more recent version of the Hubble diagram can be found in the Data Particle Book Review:

The age of the Universe

Reconsidering the equation of Friedmann for the preferred flat Universe $k = 0$ and dominated by matter ( $\Lambda = 0$ ): $H^2(t) \equiv \left(\frac{\dot R}{R}\right)^{2} = \frac{8\pi G}{3}\frac{M}{\frac{4}{3}\pi R^3 } \Rightarrow \dot{R}^2 = \frac{2GM}{R}$ , so integrating :$R^{½} dR = (2GM)^{½} dt \Rightarrow \frac{2}{3} R^{3/2} = (2GM)^{½}t \Rightarrow R(t) = \frac{3}{2} (2GM)^{1/3} t^{2/3} \Rightarrow$

$R(t) = (\frac{9GM}{4})^{1/3} t^{2/3}$ for a matter-dominated flat universe. Hence, the Hubble time, $H_0 = \frac{R(0)}{\dot{R}(0)} = \frac{3}{2}t_0 \Rightarrow t_0 = \frac{2}{3} H_0 = 9.6 \rm \, Gyr$. Other estimates provide larger values, namely $t_0 = 14 \pm 1 \rm Gyr$ (e.g. from old globular clusters) indicating or that the Universe is not flat, which is not favoured by obsevrations, or that $\Lambda \ne 0.$

A recent estimate from globular cluster indicates the value of $ t_{GC} = 13.32 \pm 0.1(stat) \pm 0.5(sys)$, which is compatible with Planck 2018 of $13.787 \pm 0.020$.

We consider again the Friedmann's equation with all contributing terms: the mass term ( $\rho_m \propto R^{-3}$ ), the radiation term ( $\rho_r \propto R^{-4}$ ), tha vacuum term ( $\rho_\Lambda = const$ ), and the curvature term with $\rho \propto \frac{1}{R^2}$ , from the equation already derived:$\Omega_{tot} = \frac{\rho}{\rho_c} = 1 + \frac{k}{R^2(0) H_0^2}$ :

$H^2(t) = H_0^2 [\Omega_m(0)(1+z)^3 + \Omega_r(0) (1+z)^4 + \Omega_\Lambda(0) + \Omega_k(0)(1+z)^2)]$

Hubble's law tells us: $H = \frac{\dot{R}}{R}$ and the redshift: $\frac{R(0)}{R(t)} = (1+z)$, then: $H = \frac{\dot{R}}{R} = -\left( \frac{\dot{z}}{(1+z)}\right) \Rightarrow dt = -\frac{dz}{(1+z)H}$

And so we obtain integrating from time t with redshift z to current time $t_0$ and $z = 0$ : $t_0 - t = \frac{1}{H_0} \int \frac{dz}{(1+z)[\Omega_m(0)(1+z)^3 + \Omega_r(0)(1+z)^4 + \Omega_\Lambda(0) + \Omega_k(0)(1+z)^2]^{1/2}}$ .

In general we can write: $t_0 = \frac{1}{H_0} F(\Omega_r, \Omega_m, \Omega_\Lambda,...)$ with F = 0.956 (from Planck 2018).

The size of the Universe for the currently observed closure parameter $\Omega_{tot} = 1$ and current preferred values are $\Omega_m(0) = 0.311 \pm 0.006, \Omega_\Lambda(0) = 0.689 \pm 0.006, \Omega_r(0) \sim 0, k = 0.$ One obtains a site of the u niverse of about $D_L \sim 3.3 \frac{c}{H_0}\sim 14.3 \rm Gpc,$ which is the comoving distance from the Earth in all directions.

The Cosmic Microwave Background

Date: 17 July 2018 Satellite: Planck Depicts: Cosmic Microwave Background Copyright: ESA/Planck Collaboration