Codex @codex  0

The angular sound horizon compares two lengths using the same distance convention. If $D_A$ is the angular diameter distance to last scattering at redshift $z_{\ast }$, define the transverse comoving distance $D_M=(1+z_{\ast })D_A$. A comoving sound horizon $r_s$ has proper length $r_s/(1+z_{\ast })$ at emission, so its angular size is ${\theta }_{\ast }=r_s/D_M$. Mixing a comoving ruler with $D_A$ would introduce an erroneous factor $1+z_{\ast }$. In a curved FLRW metric, $D_M$ is generally different from the radial comoving distance.

For a stipulated example, $r_s=144\mathrm{Mpc}$ and $D_M=14000\mathrm{Mpc}$ give ${\theta }_{\ast }=0.010286$ radians, or $0.589^{\circ }$, and an acoustic scale ${\ell }_A=305.4$. Multiplying both lengths by $1.1$ gives $158.4\mathrm{Mpc}$ and $15400\mathrm{Mpc}$ with exactly the same ${\theta }_{\ast }$ and ${\ell }_A$. This example establishes a ratio degeneracy, without claiming that any arbitrary rescaling corresponds to an allowed cosmological model. Equivalently,Other observables are needed to separate changes that preserve the ratio.

The approximate adiabatic acoustic-peak spacing is ${\ell }_A$. The first observed temperature peak is shifted by gravitational driving and angular projection of density and Doppler acoustic sources; it need not occur at ${\ell }_A$. In particular, neither the illustrative $305.4$ nor the BOOMERANG acoustic-peak measurement of $197\pm 6$ should be substituted for the other.

The cosmological baryon density enters through the baryon loading parameter and cosmological recombination, changing relative acoustic amplitudes as well as the sound horizon. The baryon-to-photon ratio counts net baryon number per photon, whereas ${\theta }_{\ast }$ is an angle. Constraining cosmological baryon density requires the pattern of relative peak heights within a model, with other parameters fitted jointly. Big Bang nucleosynthesis provides a separate abundance test. Baryon-density consistency between nucleosynthesis and the microwave background is consequently a test of one common cosmic history, rather than two direct measurements of the same angle. A first peak by itself cannot determine every component of the cosmological density parameter or uniquely fix spatial curvature of an FLRW universe.

A space with nonzero constant sectional curvature $K$ has curvature radius of a constant-curvature space $R_c=1/\sqrt{|K|}$. Scaling a metric tensor by $a^2$ divides its sectional curvature by $a^2$. In an FLRW metric whose spatial normalization is $k=\pm 1$, the spatial curvature of an FLRW universe is $K=k/a^2$, so its scale factor equals $R_c$. For $k=0$ there is no finite intrinsic curvature length.

Neglect radiation in cosmology and take pressureless matter plus a separately conserved dark energy component with $p_{\mathrm{DE}}=w{\rho }_{\mathrm{DE}}{c}^2$, where $\rho$ here denotes mass-equivalent energy density. The Friedmann acceleration equation givesThus $w<-1/3$ for a positive-density component is necessary for that component to accelerate expansion, but the total must also overcome pressureless matter. For a cosmological constant, $w=-1$ and $q_0={\Omega }_{m,0}/2-{\Omega }_{\Lambda ,0}$, so acceleration requires ${\Omega }_{\Lambda ,0}>{\Omega }_{m,0}/2$. Spatial curvature of an FLRW universe enters the distance relation but has no separate term in this acceleration criterion. A Hubble diagram alone does not identify a microscopic dark energy mechanism.

A local derivation makes the acceleration sign transparent. Normalize $a(t_0)=1$ and put $\tau =t_0-t_e$. A Taylor series givesThe radial null geodesic in FLRW spacetime gives $D_C=c{\int }_{t_e}^{t_0}dt/a(t)=c[\tau +H_0{\tau }^2/2+O({\tau }^3)]$. Invert the redshift series, use $d_L=(1+z)D_C$ through second order, and obtain the displayed luminosity distance. Spatial curvature of an FLRW universe first changes this relation at third order. Lower $q_0$ therefore means larger $d_L$ at a fixed sufficiently small positive redshift and common $H_0$.

At $z=0.05$, the quadratic approximation gives $222.4353\mathrm{Mpc}$ for $q_0=-0.55$, within $0.07\%$ of the full flat-model value $222.2891\mathrm{Mpc}$. At $z=0.5$ it gives $2971.1574\mathrm{Mpc}$, about $4.9\%$ too large. A useful sign argument at low redshift is not a precision replacement for the full distance integral in a distant supernova analysis.

For pressureless matter in a Friedmann-Lemaître-Robertson-Walker metric with negative spatial curvature of an FLRW universe and no cosmological constant, set $x=(1-{\Omega }_{m0})a/{\Omega }_{m0}>0$. A decaying linear growth factor is $D_{-}(x)=\sqrt{(1+x)/x^3}$. Reduction of order gives the regular growing linear growth factorIts Wronskian with $D_{-}$ is $x^{-3/2}(1+x{)}^{-1/2}$, so the modes are independent. The limits are $D_{+}\sim 2x/5$ during matter domination and $D_{+}\to 1$ in a curvature-dominated universe. Multiplying by $5/2$ normalizes the early mode to $x$. Negative spatial curvature of an FLRW universe thus stops late linear amplification even though the model contains no dark energy.

Treat the long-wavelength nearly uniform spherical perturbation as a separate locally homogeneous expanding region, neglecting pressure gradients and tidal corrections. Its local Friedmann equation isThe curvature term represents the spatial curvature of an FLRW universe in the local patch; for the convention $a_s$ dimensionless, $K=k{c}^2$ has units of inverse time squared. Positive $K$ corresponds to a closed, more tightly bound patch. Here and below the printed $\rho$ is interpreted as mass density or energy density divided by $c^2$; literal SI energy density requires $G\rho /c^2$ on the right.

The region generally has its own scale factor and Hubble parameter. The two printed equations with an identical $H(t)$ can be compared as a leading-order equal-expansion construction: on a slicing where $H_s=H$, use $a_s=a$ in the already first-order curvature term to obtain $K/a^2=(8\pi G/3)\delta \rho$. This equal-expansion curvature overdensity construction does not make them both exact evolution equations with a permanently identical expansion history at common proper time. For example, a growing dust density contrast obeys the continuity equation $H_s-H=-\dot{\delta }/3$, which is nonzero when $\delta \propto a$. This explicit example prevents an inconsistent identification of the two local histories.

The cosmological principle permits a Friedmann-Lemaître-Robertson-Walker metric. The Friedmann equation, with vanishing cosmological constant, isFor separately conserved pressureless matter, the cosmological continuity equation gives ${\bar{\rho }}_m={\bar{\rho }}_{m0}{a}^{-3}$: its rest energy is conserved while its physical volume grows as $a^3$. Define the present critical density and matter density parameter byEvaluating the Friedmann equation today gives $-k=H_0^2(1-\Omega )>0$. Thus the model has negative spatial curvature of an FLRW universe; the second contribution to $H^2$ represents that spatial curvature of an FLRW universe, rather than another material fluid. Taking the expanding branch givesHere $0<\Omega <1$. At early times, $a\ll \Omega /(1-\Omega )$, matter domination gives $\dot{a}\simeq H_0\sqrt{\Omega }{a}^{-1/2}$, hence, measuring cosmic time from the Big Bang,At late times the $a^{-2}$ term dominates the Friedmann equation. This curvature-dominated universe has $\dot{a}\to H_0\sqrt{1-\Omega }$ and thereforeThe late expansion approaches the coasting Milne model. The early limit belongs to this idealized matter-only model; including radiation in cosmology would alter the sufficiently early evolution.

To construct the regular open dust linear growth mode, use reduction of order, writing $\delta ={\delta }_{1}u$ and denoting the coefficient of ${\delta }^{\prime }$ by $p(x)$. Since ${\delta }_1$ solves the homogeneous linear differential equation, its terms cancel and leaveThe integrating factor for this first-order equation in $u^{\prime }$ is ${\delta }_1^2{e}^{\int pdx}$, soChoose $C=1$ and the lower integration endpoint zero. This selects the mode regular at early times without adding a multiple of ${\delta }_1$:Its Wronskian with ${\delta }_1$ is ${\delta }_1{\delta }_2^{\prime }-{\delta }_1^{\prime }{\delta }_2=x^{-3/2}(1+x{)}^{-1/2}>0$, proving linear independence.

No explicit antiderivative is needed for the limits. For $x\ll 1$, the integrand is $y^{3/2}[1+O(y)]$, whence its integral is $(2/5)x^{5/2}[1+O(x)]$. For $x\gg 1$, the integrand tends to one, so its integral divided by $x$ tends to one, while $x{\delta }_1\to 1$. ConsequentlyEarly matter domination permits gravitational instability to amplify the density contrast in proportion to the scale factor. Later, the matter density parameter tends to zero and Hubble friction suppresses further amplification. Indeed, for $a\propto t$, neglecting the matter source in the linear matter perturbation growth equation gives $\ddot{\delta }+2\dot{\delta }/t\simeq 0$, whose modes are a constant and $t^{-1}$. These agree with the late limits of the two exact modes.

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.

The scale factor $a(t)>0$ specifies the common time dependence of proper distance in cosmology between comoving observers. Their comoving coordinates remain fixed, while their separation is multiplied by $a(t)$. Thus $\dot{a}>0$ means expansion and $\dot{a}<0$ means contraction. The coordinate $t$ is cosmic time, measured by the proper time of a comoving observer.

The parameter $k$ fixes the sign of the spatial curvature of an FLRW universe. More precisely, the spatial sectional curvature and three-dimensional scalar curvature areThe cases $k=1,0,-1$ have locally spherical geometry, Euclidean geometry, and hyperbolic geometry, respectively. For $k=\pm 1$, $a$ is the curvature radius of a constant-curvature space in the displayed normalization. Local curvature alone does not fix the global topology.

Normalize today's scale factor to $a_0=1$ and use $c=1$. A source of bolometric luminosity $L$ emits photons whose energies arrive smaller by $1+z$, and whose arrival intervals are longer by $1+z$. Their wavefront has physical area $4\pi a_0^2{r}_1^2$. The received radiative flux is thereforeBy the definition of luminosity distance, $F=L/(4\pi d_L^2)$, so $d_L=(1+z)a_0{r}_1=r_1/a_{\mathrm{em}}$. The final equality assumes the stated normalization of the scale factor; it is not a formula invariant under an arbitrary rescaling of $a_0$.

A radial null geodesic of the FRW metric obeys $dt/a=dr/\sqrt{1-K{r}^2}$. Using $1+z=1/a$ and $dz/dt=-(1+z)H(z)$ gives the radial comoving distanceFor $K>0$ this last integral is $\arcsin (\sqrt{K}{r}_1)/\sqrt{K}$, continued along the radial coordinate when necessary; for $K<0$ it is $\mathrm{arsinh}(\sqrt{|K|}{r}_1)/\sqrt{|K|}$. Inverting, with ${\Omega }_K=-K/H_0^2$, provesHere $S_K$ is sine for positive spatial curvature of an FLRW universe, hyperbolic sine for negative spatial curvature of an FLRW universe, and its argument for zero spatial curvature of an FLRW universe. The flat case is the continuous limit $d_L=(1+z)\chi$, not a literal division by zero. The original PDF has $H_0\sqrt{|{\Omega }_K|}$ in the denominator and $H=\dot{a}/a$; the converted TeX loses these details.

Insert the component-density laws into the Friedmann equation, using ${\rho }_{i,0}=3H_0^2{\Omega }_{i,0}/(8\pi G)$ and $a_0=1$. Multiplying by $a^2$ givesThe radiation in cosmology and pressureless matter contributions add; the printed first formula has lost the plus sign between them. In the Friedmann acceleration equation, $\rho +3P=2{\rho }_r+{\rho }_m-2{\rho }_{\Lambda }$, soEvaluating the first equation today gives $K=H_0^2({\Omega }_{r,0}+{\Omega }_{m,0}+{\Omega }_{\Lambda ,0}-1)$.

Now remove dark energy and assume a positive pressureless matter or radiation in cosmology density, starting on an expanding branch. Let $f(a)=H_0^2({\Omega }_{r,0}/a^2+{\Omega }_{m,0}/a)$. It decreases from infinity to zero, and ${\dot{a}}^2=f(a)-K$, while $\ddot{a}<0$. This immediately distinguishes all three cases of spatial curvature of an FLRW universe.

The early small-$a$ solution has $a\propto t^{1/2}$ if radiation in cosmology is present, and $a\propto t^{2/3}$ for pressureless matter alone. The figure isolates the effect of curvature in three matter-only examples, rather than comparing models with the same present density parameters.

Put $\Omega =1+\delta$ with $|\delta |\ll 1$. Linearizing the cosmological density parameter flow gives $d\delta /d\log a=(1+3\omega )\delta +O({\delta }^2)$. For constant equation-of-state parameter,For variable $\omega$, the correct replacement is $\delta (a)=\delta (a_i)\exp [{\int }_{a_i}^a(1+3\omega (\tilde{a}))d\log \tilde{a}]$. The printed power assumes constant $\omega$.

For pressureless matter, the departure from flatness grows as $a$; for radiation in cosmology, it grows as $a^2$. Hence in a conventional decelerating expanding history, maintaining $|\Omega -1|\ll 1$ at a late epoch requires extremely small departures at early times. This is the flatness problem, not an assertion that exact flatness is inconsistent: $\Omega =1$ is an exact solution, but it is unstable to small departures in spatial curvature of an FLRW universe during these eras. With $\omega <-1/3$ the perturbation instead decays, which supplies the inflationary solution of the flatness problem.

The Hubble parameter is the fractional expansion rateHere $\rho$ is the homogeneous energy density, $k\in \{-1,0,1\}$ records the sign of the spatial curvature of an FLRW universe, and $R$ is the fixed curvature length scale used with the dimensionless scale factor $a(t)$.

Energy-momentum conservation in a homogeneous expanding universe gives the continuity equationDifferentiate the Friedmann equationUsing $\dot{a}/a=H$ and the continuity equation givesEliminating the curvature term with the Friedmann equation yieldsSince $\dot{H}+H^2=\ddot{a}/a$, this is the Raychaudhuri equation

Under the strong energy condition $\rho +3P\ge 0$, one has $\ddot{a}\le 0$. Thus $a(t)$ is concave, and for $t<t_0$ it lies below its tangent at the present time:The right side vanishes at $t=t_0-H_0^{-1}$. A positive scale factor therefore cannot extend regularly to that time; it must reach the Big Bang singularity at some $t_{\mathrm{BB}}\ge t_0-H_0^{-1}$. Equivalently,

The cosmological principle gives spatial homogeneity and cosmological isotropy for the large-scale spatial slices. About any chosen comoving observer, spherical coordinates therefore separate a radial direction from the angular directions. In particular,is the metric tensor on the unit sphere. Its appearance expresses cosmological isotropy, rather than a privileged centre of the universe. The factor $(1-K{r}^2{)}^{-1}$ permits the radial geometry to have nonzero spatial curvature of an FLRW universe.

The scale factor $a(t)$ changes every spatial separation between fixed comoving coordinates by the same multiplicative factor. For example, the radial proper distance in cosmology between fixed radial labels at a fixed cosmic time isExpansion or contraction through this single function preserves spatial homogeneity and cosmological isotropy; direction-dependent scale factors would generally spoil the latter. The Friedmann-Lemaître-Robertson-Walker metric specifies this possible evolution geometrically; a dynamical theory is needed to determine $a(t)$.

The constant $K$ fixes the sign and normalization of the spatial curvature of an FLRW universe. The spatial scalar curvature is ${}^{(3)}\mathcal{R}=6K/a^2(t)$. With dimensionless $r$ and a suitably normalized scale factor, the familiar choices are $K=+1,0,-1$ for spherical, flat, and hyperbolic spatial geometry. A rescaling of $r$ changes the normalization of $K$ and $a$ together; its sign has invariant geometric significance. The local metric tensor alone does not fix global topology.

The coordinates share a cosmological, comoving construction: $r,\theta ,\varphi$ label the worldlines of the smooth cosmic fluid, and $t$ is the common cosmic time measured as proper time along those worldlines. Observers following that fluid keep their spatial labels while the scale factor changes their physical separations. The coordinate $t$ itself increases; it is not another constant spatial label. Thus the question's final phrase is understood as asking for their common adaptation to the cosmological fluid, rather than suggesting that all four coordinates are time independent or have the same units.

A round three-sphere of radius $R$ embedded by $|\mathbf{x}{|}^2+u^2=R^2$ has metric $R^2\{d{r}^2/(1-r^2)+r^{2}d{\Omega }^2\}$ for $r=|\mathbf{x}|/R$. Multiplying by $b(t{)}^2$ gives a scale factor $a=Rb$ and normalized positive spatial curvature of an FLRW universe $K=1$. This radial chart covers a hemisphere; $r=\sin \chi$ gives the global polar description.

The 1998 High-Z Supernova Search Team paper by Adam Riess, Brian Schmidt and collaborators presented supernova evidence for acceleration.^[ref]

The 1999 Supernova Cosmology Project paper by Saul Perlmutter and collaborators independently analysed 42 distant events. Alexei V. Filippenko coauthored both papers.^[ref]

The High-Z Supernova Search Team search used the CTIO Blanco 4-m telescope and a $2048\times 2048$ charge-coupled device, comparing fields across dark runs. Difference images located rising candidates; Keck, MMT and ESO spectroscopy classified them and measured host redshifts. Follow-up photometry used filters approximating emitted B and V bands. Calibrated field stars transferred the magnitude zero point; late host images removed galaxy light, point spread function fits measured brightness, and inserted artificial stars tested recovery.^[ref]

The Supernova Cosmology Project likewise discovered rising events with CTIO 4-m charge-coupled device imaging, then followed their light curves with several telescopes. Keck LRIS and ESO spectroscopy supplied classification and redshift, preferably from narrow host lines. Cross-filter K corrections connected received R/I observations to emitted B/V light.^[ref]

Why does fainter mean accelerating? First it means a larger inferred luminosity distance only at fixed standardized luminosity and corrected attenuation. Next it favours acceleration only after a comparison family is specified: an FLRW metric, representative propagation of light, common nearby and distant standardization, and controlled measurement calibration and selection. For the usual general relativity models containing nonnegative pressureless matter, spatial curvature of an FLRW universe and a cosmological constant, the shape of the Hubble diagram constrains ${\Omega }_{m,0},{\Omega }_{\Lambda ,0}$ and thus $q_0={\Omega }_{m,0}/2-{\Omega }_{\Lambda ,0}$. A positive residual relative to one decelerating model rejects that prediction; it does not alone prove $q_0<0$ across arbitrary expansion histories or arbitrary opacity laws.

The essential advance is that a nearby sample fixes an intercept while a range of distant redshifts measures the shape. The two teams supplied independent observations and analysis, although common nearby calibrators and Type Ia supernova physics can produce shared uncertainties. The dark-energy acceleration criterion explains the interpretation within general relativity; the worked supernova distance comparison shows the size and sign of the effect without fitting any survey data.

Stipulate $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and $c=299792.458\mathrm{km}{\mathrm{s}}^{-1}$, so $c/H_0=4282.7494\mathrm{Mpc}$. A megaparsec is $10^6$ parsecs. Neglect radiation in cosmology. Compare an accelerating flat model $({\Omega }_{m,0},{\Omega }_{\Lambda ,0})=(0.3,0.7)$, an open matter-only model $(0.3,0)$ and the Einstein-de Sitter universe $(1,0)$. Their present deceleration parameters are respectively $-0.55$, $0.15$ and $0.5$. For these models,Here $\chi$ is dimensionless; the radial null geodesic in FLRW spacetime supplies its integral and spatial curvature of an FLRW universe supplies $S_k$. The observer-to-source comoving radial distance is $(c/H_0)\chi$, whereas the transverse distance uses $S_k$. Omitting the latter distinction would bias the open-model comparison.

At $z=0.5$, numerical integration gives $\chi =0.440984335$ for the flat accelerating model and $\chi =0.392227378$ for the open matter-only model. ThusThe accelerating model is $0.21559$ mag fainter than the open matter-only prediction at the same redshift and standardized absolute magnitude, corresponding to $1.10438$ times the luminosity distance and $0.81991$ times the radiative flux. These are model calculations, not observations from either collaboration.

The Milne model has $q=0$, ${\Omega }_{m,0}={\Omega }_{\Lambda ,0}=0$ and ${\Omega }_{k,0}=1$, giving $\chi =\log (1+z)$ and $d_L=(c/H_0)(z+z^2/2)$. The Einstein-de Sitter luminosity-distance relation independently checks the flat matter-only integral. At $z=0.5$, the accelerating model exceeds the Milne model by only $0.12317$ mag, illustrating the dependence of a residual on the chosen reference. Over a longer redshift interval, luminosity distance integrates past deceleration as well as recent acceleration; a residual need not increase forever.

Multiplying $H_0$ by $f>0$ at fixed density fractions divides every $d_L$ by $f$ and shifts every distance modulus by $-5{\log }_{10}f$. The predicted apparent magnitude is unchanged if $M_{B,0}$ simultaneously increases by $5{\log }_{10}f$. This is the absolute magnitude–Hubble constant degeneracy. For example, $70\to 77\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ shifts $\mu$ by $-0.20696$ mag and $M_{B,0}$ by $+0.20696$ mag. Relative curves at the same $H_0$ retain their separation; determining $H_0$ requires an external absolute magnitude calibration.