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Match flat Friedmann equations histories at scale factor $R_1$ and Hubble parameter $H_1$, and give them identical subsequent evolution. Below $R_1$, constant vacuum energy gives $H=H_1$, while a radiation-dominated universe gives $H=H_1(R_1/R{)}^2$. From $ϵR_1$ to $R_1$ the former takes longer by $H_1^{-1}{\int }_{ϵ}^1(1/x-x)dx>0$. Its duration diverges as $ϵ\to 0$; the radiation duration tends to $1/(2H_1)$. A finite period of cosmic inflation requires a finite initial state and a specified matching before such age comparisons are meaningful.

In an FLRW metric with zero separate cosmological constant, $\varrho +3p\ge 0$ makes the Friedmann acceleration equation imply $\ddot{a}\le 0$. If $a_0=a(t_0)>0$ and $v_0=\dot{a}(t_0)>0$, then the scale factor is a concave function andthroughout its regular past interval. Thus a regular positive scale factor cannot persist earlier than $t_0-a_0/v_0$. If the solution continues regularly whenever $a>0$, its first past zero occurs within $a_0/v_0=H(t_0{)}^{-1}$, where $H$ is the Hubble parameter.

The continuation condition matters: without restrictions on the equation of state, an earlier curvature singularity can occur at positive $a$. For example, in fixed units $a(t)=1+\sqrt{t}$, $t>0$, with $k=0$, defines energy density $\varrho =3(\dot{a}/a{)}^2/(8\pi G)$ and pressure $p=-\varrho /3-\ddot{a}/(4\pi Ga)$. These satisfy the Friedmann equations and $\varrho +3p>0$, yet $a\to 1$ at the singular past boundary. Conversely the Milne model reaches zero without divergent curvature. A finite duration, a zero of the scale factor, and a curvature singularity are distinct conclusions.

For a spatially flat universe consisting only of dust and constant cosmological constant, $A_0={\ddot{a}}_0/a_0$ and $H_0$ determine the two model parameters through the Friedmann equations. Predicting the future also assumes those matter laws persist. Local derivatives alone do not determine arbitrary future stress evolution: $a_{+}(t)=1+t+t^2/2$ and $a_{-}(t)=1+t+t^2/2-t^3$ agree through second derivative at zero, but the first expands forever and the second recollapses.

The lecture notes by David Tong derive Friedmann equations in section 4.6.2 and classify Reissner-Nordstrom spacetime in sections 6.2.1–6.2.5. Section 6.2 distinguishes the central curvature singularity from regular horizons, including the extremal limit.

For conserved nonrelativistic dark matter and radiation, ${\rho }_{\mathrm{DM}}\propto a^{-3}$ and ${\rho }_{\gamma }\propto a^{-4}$, so $y={\rho }_{\mathrm{DM}}/{\rho }_{\gamma }=a/a_{\mathrm{eq}}$ obeys $\dot{y}=yH$. Denote differentiation with respect to $y$ by a prime. ThenThe supplied Friedmann equations implySubstitution in the linear growth equation and division by $y^2{H}^2$ give the Mészáros equationFor $D_1=1+3y/2$, $D_1^{\prime }=3/2$ and $D_1^{\prime \prime }=0$, making the two remaining terms cancel exactly. For reduction of order, put $\delta =f{D}_1$ and $P=(2+3y)/[2y(1+y)]=1/y+1/[2(1+y)]$. The equation reduces to $f^{\prime \prime }+(2D_1^{\prime }/D_1+P)f^{\prime }=0$, whenceUsing the supplied antiderivative and reversing an irrelevant overall sign gives the independent solutionIts Wronskian is $D_1{D}_2^{\prime }-D_1^{\prime }{D}_2=-1/[y\sqrt{1+y}]\ne 0$. The asymptotic expansions areThus subhorizon dark matter during radiation domination has constant and logarithmic modes, rather than the superhorizon $a^2$ growth of b. A combination with negative $A_2$ supplies positive logarithmic growth. During matter domination, the growing mode is proportional to $a$, agreeing with b, while the second mode decays as $a^{-3/2}$. These results neglect subhorizon radiation clustering; the Mészáros equation is the approximation stated in the question, not a full photon–matter perturbation system through horizon entry.

The official notes, PDF pp. 17–18, handwritten pp. 10–11 give the matter density, curvature and cosmological constant scalings and the requested logarithmic sketch. For the closed model the curvature contribution to the Friedmann equations is negative; the sketched curvature magnitude must be read with that sign restored.

The official notes, PDF pp. 20–21, handwritten pp. 13–14 combine the Friedmann equations and Raychaudhuri equation to derive the density and scale factor of the Einstein static universe.

The official notes, PDF pp. 22–23, handwritten pp. 15–16 give both density scalings, integrate the radiation- and matter-dominated Friedmann equations, normalize the late solution today, and match the two approximations at equality.

Use standardized Type Ia supernovae as distance indicators and measure their redshifts. The curvature of the luminosity-distance relation constrains the deceleration parameter:Fainter supernovae at a given redshift imply larger distances, and comparison with the predicted curve tests accelerated expansion. In practice one fits the full expansion history over the observed redshift range and controls luminosity evolution, extinction and calibration, rather than relying only on a short low-$z$ series.

The cosmological fate from local expansion data is conditional on a matter model. For the spatially flat model of part b, if the only relevant components are pressureless matter and a constant $\Lambda$, put $A_0=\ddot{R}(t_0)/R(t_0)$. The two Friedmann equations giveHere $\Lambda$ follows the sign of the supplied Friedmann equations, where positive $\Lambda$ contributes positive acceleration. These fix the parameters governing subsequent evolution. Positive $\Lambda$ yields eternal late-time accelerated expansion; zero $\Lambda$ gives the flat matter solution; negative $\Lambda$ leads to eventual recollapse in this dust model. If instead one assumes dust and zero $\Lambda$ while allowing curvature, $q_0={\Omega }_{m0}/2$ determines whether the matter universe is closed and recollapsing or expands indefinitely.

Without a fixed equation of state and its future evolution, two present derivatives do not uniquely fix the future. For a concrete counterexample in units with $t_0=0$, the positive scale factors $a_{+}(t)=1+t+t^2/2$ and $a_{-}(t)=1+t+t^2/2-t^3$ have the same $a,\dot{a},\ddot{a}$ now. The former expands forever; the latter turns around and later reaches zero. Different time-dependent matter stresses can support their different flat expansion histories. Thus the printed sufficiency statement needs the cosmological-model assumption.

On large scales the appropriate background is the spatially flat FLRW metric,Here $t$ is cosmic time, the proper time of a comoving observer; $R(t)$ is the scale factor; and $(r,\theta ,\varphi )$ are dimensionless comoving coordinates. With this convention $R$ has units of length. A fixed comoving radial interval has instantaneous proper length $R(t)dr$. The cosmological principle assumes spatial homogeneity and cosmological isotropy on the scales where this background description applies.

The general FLRW metric instead has $d{r}^2/(1-k{r}^2)$ in its radial part, with normalized $k=+1,0,-1$ describing positive, zero or negative spatial curvature of an FLRW universe. The best simple description takes $k=0$; observations constrain spatial curvature to be small rather than proving it identically zero. The late accelerated expansion is encoded by the time dependence of $R$, determined by the stress-energy tensor through the Friedmann equations, rather than by an additional term in the FLRW metric. Small-scale galaxies and other inhomogeneities require perturbations of this background.

For $a>0$, the assumed inequality and the Friedmann acceleration equation imply $\ddot{a}\le 0$. Thus $a$ is a concave function. Let $a_0=a(t_0)>0$ and $v_0=\dot{a}(t_0)>0$. At every earlier time in the same regular expanding interval, $\dot{a}(t)\ge v_0$. Integration gives the concavity bound on past FLRW evolution:The right-hand side reaches zero at $t=t_0-a_0/v_0$. Hence an everywhere regular positive scale factor cannot extend arbitrarily far into the past. With the usual implicit continuation assumption that the cosmological solution remains regular while $a>0$, it reaches $a(t_B)=0$ at a finite past boundary satisfyingPast expansion lasts at most the present inverse Hubble parameter. The inequality is the essential argument, and equality is attained by a linearly growing scale factor.

There is a literal qualification to the claim that $a$ must reach zero: the printed hypothesis alone does not exclude an earlier curvature singularity with positive limiting $a$. In arbitrary fixed units, take $k=0$ and $a(t)=1+\sqrt{t}$ for $t>0$, and define $\varrho ,p$ using the two Friedmann equations and the Friedmann acceleration equation. Then $\dot{a}>0$, $\ddot{a}<0$, and $\varrho +3p=-3\ddot{a}/(4\pi Ga)>0$, but $a\to 1$ while the energy density and pressure diverge as $t\downarrow 0$. Thus the rigorous unconditional conclusion is the finite bound on a regular past interval; reaching zero additionally requires the stated continuation assumption. Nor does a zero of $a$ alone prove a curvature singularity, as the Milne model illustrates.

Each separately conserved component obeys the constant-equation-of-state density scaling $\rho \propto a^{-3(1+w)}$. For the flat Friedmann-Lemaître-Robertson-Walker metric, the Friedmann equations therefore giveWrite $H(z)/H_0=1+h_{1}z+O(z^2)$, where $h_1=[3{\Omega }_m+4{\Omega }_r+3(1+w){\Omega }_X]/2$. Expanding the reciprocal inside the luminosity distance integral givesMatching its quadratic coefficient gives the deceleration parameter $q_0=h_1-1$. Using flatness in this part,A negative deceleration parameter signifies accelerated expansion; negative pressure must overcome the positive matter and radiation terms. Restoring units replaces $H_0{d}_L$ by $H_0{d}_L/c$ in the expansion, leaving $q_0$ unchanged.

There are two normalization problems in the printed energy identity. In conformal time, the physical density of a homogeneous scalar is $T_{00}/a^2$, and its kinetic contribution contains ${\varphi }^{\prime 2}$, not one power of ${\varphi }^{\prime }$. The conformal density of a canonical scalar field follows directly from the stress-energy tensor:Writing $K={\varphi }^{\prime 2}/(2a^2)$, the condition $K-V=-(K+V)/3$ gives $V=2K$, henceThe reconstruction is consistent with this corrected identity; the unsquared printed identity cannot hold as an energy relation.

The constant-equation-of-state density scaling for this coasting fluid gives ${\rho }_{\varphi }={\rho }_{X0}{a}^{-2}$, where ${\rho }_{X0}=3M^2{H}_0^2{\Omega }_X$. Choose the branch on which $\varphi$ increases. ThenKeeping all three density parameters explicitly, the Friedmann equations in conformal time giveFor ${\Omega }_X>0$, defineThe square root is $\sqrt{(a+b{)}^2-A^2}$. Integrating on the expanding branch giveswhere $a({\varphi }_0)=1$. Thus the scalar-field reconstruction of a coasting component isSubstitution into the potential gives the requested present-field parametrization:The domain is the branch where $a>0$ and the hyperbolic argument is positive; reversing the direction of the field changes the sign of $B$. The actual discriminant condition is ${\Omega }_m^2/4>{\Omega }_r{\Omega }_X$. The supplied stronger bound suffices for physical density fractions $0<{\Omega }_X\le 1$. No density parameter has been eliminated using the flatness sum.

As a check, ${\varphi }^{\prime }$ is constant and $V_{,\varphi }=-2V(a^{\prime }/a)/{\varphi }^{\prime }$. Since $a^{2}V={\varphi }^{\prime 2}$ along this solution, ${\varphi }^{\prime \prime }+2(a^{\prime }/a){\varphi }^{\prime }+a^2{V}_{,\varphi }=0$ identically. The reconstructed field therefore satisfies its equation of motion as well as the Friedmann equations and required equation of state.

Type Ia supernovae have similar peak luminosities that can be standardized using their light-curve width and color, rather than assumed to be identical without correction. Their redshifts and corrected apparent magnitudes give a distance modulus,Compare this Hubble diagram with the luminosity-distance prediction of a background cosmology. Restoring $c$, put $E(z)=H(z)/H_0$, $I(z)={\int }_0^{z}d{z}^{\prime }/E(z^{\prime })$ and define ${\mathcal{S}}_{{\Omega }_k}(I)$ as $\sinh (\sqrt{{\Omega }_k}I)/\sqrt{{\Omega }_k}$ for ${\Omega }_k>0$, $I$ for ${\Omega }_k=0$, and $\sin (\sqrt{-{\Omega }_k}I)/\sqrt{-{\Omega }_k}$ for ${\Omega }_k<0$. ThenNeglecting radiation at these redshifts, Friedmann equations and a separately conserved dark energy component with $w(z)=p/(\rho c^2)$ giveFor constant $w$, the last factor is $(1+z{)}^{3(1+w)}$; a cosmological constant has $w=-1$. Thus the redshift dependence of the measured distance constrains matter density, curvature and the dark-energy equation of state. At small redshift, $d_L=(cz/H_0)[1+(1-q_0)z/2+\cdots ]$, showing how departures from the linear Hubble law probe the deceleration parameter.

The early distant-supernova analyses found larger distances than matter-only decelerating models predicted, providing evidence for accelerated expansion; see the 1998 supernova analysis and the 1999 independent supernova analysis. This is a statement about the distance-redshift curve under the standardization and cosmological assumptions, not a direct measurement of a fluid pressure.

The absolute magnitude–Hubble constant degeneracy means uncalibrated supernovae constrain the shape of $d_L(z)$ without separately determining $H_0$ and the common absolute luminosity. The integral over $H(z)$ also makes different matter, curvature and $w(z)$ histories partly degenerate. Wide redshift coverage and independent cosmic microwave background and large-scale structure of the universe information help break these degeneracies. Dust extinction, source evolution, photometric calibration, sample selection and K corrections must be modeled, and constraints on a time-dependent equation of state depend on how that function is parameterized. Standardized supernovae test the expansion history; they do not by themselves uniquely determine every cosmological parameter.

Take zero cosmological constant and an open geometry, which means $k<0$ in the metric convention. Evaluating the Friedmann equations today gives $H_0^2=H_0^2{\Omega }_0-k/a_0^2$, soOn a constant-time slice the radial line element is $d\ell =a_{0}dr/\sqrt{1-k{r}^2}$. ThereforeThis is a cosmological proper distance on today's slice, not the distance traveled during the photon flight. The printed condition $k<1$ is insufficient to specify an open model; the derivation uses its intended $k<0$.

Physical cosmology connects the universe's history to the measurable behaviour of its contents. The Friedmann equations describe cosmic expansion; atomic physics, radiation pressure and gravity determine what that expanding universe looks like. A particularly revealing example is the route from cosmological recombination through the photon-baryon fluid to the Cosmic microwave background acoustic peaks. The universe supplies the initial conditions and the laboratory; the prediction must still pass through a model of the telescope before it can be compared with observations.

The practical reward is that one sky contains several different tests. The Cosmic microwave background power spectrum carries information about the sound horizon, the cosmological baryon density and cosmic geometry. These are related quantities with different observational signatures. Angular sound-horizon degeneracy explains why measuring a characteristic angle alone cannot count the universe's baryons.

The scale factor $a(t)>0$ measures how spatial lengths in an FLRW metric change with cosmic time $t$. Choose $a$ to be a dimensionless quantity with $a(t_0)=1$ at a reference epoch, usually today. For two comoving observers, their fixed radial comoving distance $\chi$ and their proper distance in cosmology $d(t)$ satisfyThe proper distance in cosmology is measured along a spatial slice at one cosmic time; it is not the distance that a travelling photon covers between emission and reception. Only ratios of the scale factor have a normalization-independent meaning: replacing $a$ by $Ca$ and $\chi$ by $\chi /C$ leaves $d$ unchanged.

For example, take $\chi =100\mathrm{Mpc}$, where $\mathrm{Mpc}$ means a million parsecs. At $a=0.5,1,1.5$, the same pair of comoving observers has proper distance in cosmology $50,100,150\mathrm{Mpc}$ respectively. Every such separation changes by the same factor, although more distant pairs gain more distance. The illustration treats the marked objects as ideal comoving observers; internal motions in galaxies need not follow this background scaling.

The beauty of the scale factor is that one function describes the changing spatial ruler for every comoving observer. Spatial homogeneity gives no preferred centre of cosmic expansion: in a flat spatial patch, relative positions obey $\mathbf{r}-{\mathbf{r}}_O=a(\mathbf{x}-{\mathbf{x}}_O)$, so recentering on any comoving observer preserves Hubble's law. The evolution of the scale factor is determined by the Friedmann equations, while its ratio between emission and reception determines cosmological redshift.

In a spatially flat universe dominated by a positive constant energy density $ϵ$, the Friedmann equations give $H^2=8\pi Gϵ/(3c^2)$. On the expanding branch $a(t)=a_i{e}^{H(t-t_i)}$, so the scale factor doubles after $(\log 2)/H$. This concerns a linear length scale; the enclosed physical volume grows by a factor of eight over the same interval.