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A kinematic redshift is the relativistic Doppler effect arising from relative motion of emitter and receiver. In a local inertial frame, recession decreases the received frequency and approach increases it. A cosmological redshift instead compares photons emitted and received by comoving observers at different cosmic times: expansion of the scale factor stretches their wavelengths. It exists even when both observers have zero peculiar velocity. Both are frequency ratios measured by observers, but a large cosmological redshift should not be inserted into a flat-spacetime radial-speed formula to define the galaxy's unique recession speed.

In the intended comparison of emission and reception epochs, with negligible peculiar velocities, the Earth-observed cosmological redshifts correspond to $a_A=a_0/2$ and $a_B=a_0/10$. A photon emitted at the B epoch and received at the A epoch consequently hasThese factors divide, whereas the redshifts themselves do not subtract. There is a geometrical qualification to this usual answer: the two Earth-observed redshifts alone do not specify an arbitrary actual B-to-A observation. The assumed events must be connected by a null geodesic, as when A is on the same radial photon path between B and Earth. If the galaxies have an unspecified angular separation, or A observes at a different cosmic time, the B emission event reaching A is different and its redshift is not determined by these two numbers alone.

The neutral-hydrogen column density is $N_{\mathrm{HI}}=\int n_{\mathrm{HI}}d\ell$, with units ${\mathrm{cm}}^{-2}$. Define the neutral-hydrogen column-density distribution byHere $\mathcal{N}$ counts absorption systems along a quasar sightline. A histogram per logarithmic column interval instead measures $N\ln (10)f(N,z)$; forgetting this factor changes the inferred slope. Another common convention uses the absorption distance $X$, with $dX/dz=H_0(1+z{)}^2/H(z)$, and $f_X=f/(dX/dz)$.

At a fixed epoch, weak Lyman-alpha forest lines have an approximately falling power-law distribution, $f(N,z)\simeq A(z)N^{-\beta }$ with $\beta \simeq 1.4$–$1.5$ over representative low-column ranges. Stronger absorbers deviate from one universal slope, and the normalization and fitted slopes depend on epoch and sample. The low-column fit is measured, for example, in Kim and collaborators' forest survey. The sketch distinguishes that empirical low-column trend from an illustrative higher-column continuation.

Ordinary Lyman-alpha forest lines cover roughly $10^{12}\lesssim N_{\mathrm{HI}}\lesssim 10^{17.2}{\mathrm{cm}}^{-2}$. They are numerous resonant Lyman-alpha absorption features from intervening gas at different cosmological redshifts, collectively resembling a forest in a quasar spectrum. Many are optically thin at the ionization edge even when their line centers are saturated.

A Lyman limit system becomes optically thick at the $91.2\mathrm{nm}$ hydrogen ionization edge. With ${\sigma }_{912}\simeq 6.3\times 10^{-18}{\mathrm{cm}}^2$, ${\tau }_{912}=N_{\mathrm{HI}}{\sigma }_{912}\gtrsim 1$ requiresIts spectrum consequently has a strong continuum decrement below the redshifted Lyman edge. In a classification separating the highest columns, Lyman limit systems extend up to the damped-system boundary.

A damped Lyman-alpha system conventionally has $N_{\mathrm{HI}}\gtrsim 2\times 10^{20}{\mathrm{cm}}^{-2}$. Its Lyman-alpha absorption is sufficiently strong that the Lorentzian wings of the Voigt profile are prominent. “Damped” refers to the finite radiative lifetime of the atomic excited state, not an inference that the gas has a high temperature. These systems trace substantial neutral reservoirs, usually associated with galaxies.

For pressureless matter, the cosmological perfect-fluid continuity equation gives $\rho \propto a^{-3}$. The flat Friedmann equation then gives $a\propto t^{2/3}$, and conformal time $d\eta =dt/a$ gives $a\propto {\eta }^2$. Choose the initial conformal time to be zero. A galaxy seen at emission time ${\eta }_e$ has comoving radius $\chi ={\eta }_0-{\eta }_e$ and cosmological redshift $1+z=({\eta }_0/{\eta }_e{)}^2$.

Two opposite comoving galaxies are separated by $2\chi$. Direct one-way communication between their worldlines before emission requires the particle horizon ${\eta }_e$ to cover this separation: $2({\eta }_0-{\eta }_e)\le {\eta }_e$. Thus the threshold is ${\eta }_e=2{\eta }_0/3$, andThe distinction in direct cosmological communication and a shared causal past matters here. If causal contact means that their past light cones overlap, the condition instead is $2\chi \le 2{\eta }_e$, giving ${\eta }_e={\eta }_0/2$ and $\boxed{z_c=3\text{ for a shared causal past}}$. These are different operational meanings of the horizon criterion; the boundary corresponds to the limiting initial-time contact.

A radial null geodesic in the FLRW metric obeys $d\chi =c|dt|/a$. The redshift-time relation $dt=-dz/[(1+z)H(z)]$, with $a_0=1$, therefore givesThis is comoving radial distance. The proper distance in cosmology at the clouds' approximately common epoch is insteadThe distinction is essential when quoting a cloud separation. The approximation neglects the change of $H$ across the interval and assumes the observed spacing is dominated by cosmological redshift; peculiar velocity can perturb the inferred separation. It is the redshift version of proper absorber separation from a wavelength interval.

The Lyman-alpha forest is the collection of absorption lines blueward of a distant quasar's own Lyman-alpha line. Foreground neutral hydrogen removes photons at rest wavelength ${\lambda }_{\alpha }\simeq 1215.67$ angstroms, observed at $\lambda =(1+z){\lambda }_{\alpha }$ because of cosmological redshift. Gas at many different foreground redshifts therefore produces many lines. The weak forest traces diffuse, mostly ionized intergalactic medium, rather than a population of fully neutral clouds; much stronger, self-shielded absorbers are a separate regime. Madau's account of quasar absorption systems distinguishes these column-density regimes.

The comoving particle horizon is the maximum radial distance a photon could have travelled since the initial Big Bang, expressed in comoving coordinates. It describes possible past causal contact, whereas a cosmological event horizon concerns photons that can reach an observer in the future.

For a radial null geodesic in the FLRW metric, $dr/\sqrt{1-k{r}^2}=cdt/a$. Spatial flatness sets $k=0$, so with $a_0=1$,The given Hubble parameter then supplies $a^{2}H=H_0\sqrt{{\Omega }_{r,0}+{\Omega }_{m,0}a+{\Omega }_{\Lambda ,0}{a}^4}$. Since cosmological redshift obeys $a=(1+z{)}^{-1}$,The corresponding proper horizon at that epoch is $a(z)r_p(z)$.

The photon number density is obtained by integrating the Planck photon distribution. With $x=h\nu /(kT)$,The integral converges, since its integrand is of order $x$ near zero and decays exponentially at infinity. Hence $n=\alpha T^3$, with $\alpha$ independent of temperature.

For equilibrium radiation undergoing adiabatic expansion, the total entropy in a comoving volume is constant. Since the entropy density is $\beta T^3$, this means $\beta T^{3}V$ is constant. A comoving volume scales as the cube of the scale factor, soThis agrees with the cosmological redshift of each photon frequency. The conclusion uses entropy conservation during the expansion.

For neighbouring comoving observers separated by a small physical distance $d\ell$, the recession velocity is $H(t)d\ell$, where $H=\dot{a}/a$ is the Hubble parameter. During the photon's travel time $dt=d\ell /c$, the local Doppler effect gives $d\lambda /\lambda =Hdt$. Integrating along the ray gives the cosmological wavelength-redshift relation:The same ratio stretches the arrival interval of successive wave crests and of successive emitted pulses: $d{t}_0=(1+z)d{t}_e$. Each photon loses a factor $1+z$ in energy because $E=hc/\lambda$. Thus energy arriving through the sphere per unit observer time has one factor from cosmological redshift and one from time dilation:This is the total power crossing the sphere; division by its area would give the observed flux.

Let $X$ be the hydrogen mass fraction, $f_g={\rho }_g/{\rho }_{\mathrm{tot}}$ the gravitating gas fraction, and $\chi =n_{\mathrm{tot}}/n_H$ the total-particle-to-hydrogen ratio. The optically thin gas cooling time and free-fall time of a uniform sphere arewhere ${\rho }_{\mathrm{tot}}=m_p{n}_H/(X{f}_g)$. Equating them gives the cooling-to-free-fall equality curve:At fixed composition, its shape is $n_{\mathrm{crit}}\propto T^2/{\Lambda }^2$. Gas above this curve cools faster than it falls; gas below it cools more slowly. Strong line cooling produces a deep low-density trough. In the high-temperature free-free regime, $\Lambda \propto T^{1/2}$ gives $n_{\mathrm{crit}}\propto T$.

The displayed uniform self-gravitating-cloud example uses $f_g=1$, $X=0.76$, $\mu \simeq 0.59$ and $\chi \simeq 2.24$. Holding these ionized-gas factors fixed makes the diagram transparent, but near the atomic threshold their actual temperature dependence changes numerical normalizations. With these conventions the supplied cooling values imply $n_{\mathrm{crit}}(1.5\times 10^4\mathrm{K})\simeq 9.6\times 10^{-10}\mathrm{c}{\mathrm{m}}^{-3}$ and $n_{\mathrm{crit}}(10^7\mathrm{K})\simeq 0.107\mathrm{c}{\mathrm{m}}^{-3}$. For gas supported in a dark-matter-dominated dark matter halo, take $f_g<1$: at fixed $n_H$ the free-fall time of a uniform sphere is shorter and the critical curve shifts upward by $1/f_g$.

To add virial mass contours in a cooling diagram, use the uniform self-gravitating sphere consistently. Its Newtonian gravitational potential energy is $-3G{M}^2/(5R)$ and its thermal kinetic energy is $(3/2)M{k}_{B}T/(\mu m_p)$. The virial theorem gives $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(5R)$, while $M=(4\pi /3)R^3{\rho }_{\mathrm{tot}}$. Eliminating $R$ givesThus constant total gravitating mass gives slope three in the logarithmic density-temperature plane, with larger mass contours lying at smaller density. A different density profile changes the structural coefficient; the customary halo temperature scale $k_B{T}_{\mathrm{vir}}=\mu m_{p}GM/(2R)$ shifts the mass contours but preserves their slopes. The horizontal illustrative formation-density lines are $n_{H,\mathrm{vir}}\simeq {\Delta }_v(2\times 10^{-7})(1+z{)}^3\mathrm{c}{\mathrm{m}}^{-3}$, using ${\Delta }_v=18{\pi }^2$. They show why increasing formation cosmological redshift helps gas reach the rapid-cooling region.

At the low-mass end, a dark matter halo's virial temperature may not reach the atomic threshold. The atomic and molecular cooling thresholds for galaxy formation therefore favor an atomic-cooling mass of order $10^8{M}_{\odot }$ at early formation epochs, with its precise value varying as approximately $(1+z{)}^{-3/2}$. Smaller primordial systems require molecules; photoheating and stellar feedback further suppress their luminous output.

At the high-mass end, the virial temperature at a fixed assembly density scales as $M^{2/3}$. Beyond the line-cooling interval the gas becomes increasingly hot, while thermal bremsstrahlung improves only as $T^{1/2}$. Hence at fixed assembly density $t_{\mathrm{cool}}/t_{\mathrm{ff}}\propto T^{1/2}\propto M^{1/3}$ in that regime. Large systems can remain pressure-supported hot atmospheres for many dynamical times, instead of promptly fragmenting into one stellar object. This is the upper galaxy mass from gas cooling: it associates efficient condensation with roughly galactic, rather than group or cluster, masses and motivates the broad $10^8$–$10^{12}{M}_{\odot }$ interval.

The diagram gives characteristic cooling scales, not exact universal mass bounds. Density profiles, gas fraction, enrichment and formation epoch move the boundaries; the two supplied values alone do not determine a complete numerical cutoff. Some gas can cool in dense cores of larger dark-matter haloes, early smaller progenitors can form stars before merging, and subsequent galaxy mergers can assemble more massive galaxies. Long-lived hot atmospheres also require heating to prevent excessive late-time cooling. These qualifications explain why the cooling criterion for galaxy formation is useful without equating each dark dark matter halo to one galaxy.

For an optically thin low-density gas in collisional ionization equilibrium, the astrophysical cooling function collects losses per pair of hydrogen nuclei into $\mathcal{C}=n_H^2\Lambda (T)$. Its units are $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$; the time unit is missing from the printed guide to the vertical-axis labels. For the primordial atomic cooling curve, only hydrogen and helium provide bound-state coolants.

Near $10^4\mathrm{K}$, collisional excitation of hydrogen becomes effective and subsequent photon emission removes thermal energy. The excitation rate is exponentially suppressed below the atomic energy threshold. Atomic line cooling and collisional ionization generate a strong hydrogen feature around a few $10^4\mathrm{K}$; helium excitation and ionization produce further structure around $10^5\mathrm{K}$. Radiative recombination also contributes. Once the gas is highly ionized, these bound-state losses weaken. At $T\gtrsim 10^6$–$10^7\mathrm{K}$, thermal bremsstrahlung dominates, with approximately $\Lambda \propto T^{1/2}$ apart from slowly varying factors. The figure is an original qualitative sketch anchored to the two supplied values, not an atomic-rate calculation.

Increasing galactic metallicity adds many ionic transitions, producing metal-line cooling and substantially enhancing cooling, particularly in the $10^5$–$10^7\mathrm{K}$ range. Metals also provide low-energy fine-structure transitions below $10^4\mathrm{K}$. The change is not a uniform vertical shift: the positions and strengths of features depend on the elemental abundances and ionization state.

Photoionization removes bound electrons even where collisions alone would leave atoms neutral, often suppressing the hydrogen and helium line peaks and changing the metal-ion population. It also adds photoionization heating. The net thermal loss is then $\mathcal{C}-\mathcal{H}$, with possible heating-cooling equilibrium near $10^4\mathrm{K}$. A single density-independent $\Lambda (T)$ no longer describes every irradiated cloud: the answer also depends on radiation intensity, spectrum, shielding and density. The illustrative photoionized curve denotes altered cooling alone, not the net loss including heating.

Below $10^4\mathrm{K}$, primordial gas can use molecular line cooling, especially rotational and vibrational transitions of molecular hydrogen and, in suitable chemical conditions, hydrogen deuteride. Molecule formation and protection against photodissociation are essential. Enriched gas can additionally use metal-line cooling, other molecular species and dust thermal emission, with gas-dust energy exchange important at high density. At high cosmological redshift, residual free electrons can transfer energy to the cosmic microwave background through Compton cooling by the cosmic microwave background if the gas is hotter than the radiation. Expansion can also cool gas adiabatically, but neither that process nor Compton cooling by the cosmic microwave background is an $n_H^2$ atomic loss law. The atomic threshold is a cooling-channel limitation, not a universal minimum gas temperature.

A scale-invariant primordial potential has $P_{\Phi }(k)\propto k^{-3}$. On subhorizon linear scales, the cosmological Poisson equation and the cold-dark-matter transfer function give ${\delta }_c(k,a)\propto D(a)k^{2}T(k){\Phi }_{\mathrm{prim}}(k)$, where $D(a)\propto a$ during matter domination. ConsequentlyLarge scales enter only after equality and have $T(k)\simeq 1$. Small scales enter during radiation domination; their approximately logarithmic growth up to equality gives $T(k)\propto (k_{\mathrm{eq}}/k{)}^2\ln (k/k_{\mathrm{eq}})$. ThereforeIf the slow logarithm is suppressed in a rough sketch, the slopes are $+1$ and $-3$, with a turnover near $k_{\mathrm{eq}}$. The logarithmic correction is real and should not be mistaken for a different primordial spectral index. In the matter-only late-time approximation with $D(0)=1$, the cosmological redshift $z=1$ means $a=1/2$ and $D(1{)}^2=1/4$; it changes the amplitude, not these asymptotic shapes. Neither the initial normalization nor cosmological parameters needed for an absolute power are supplied.

This is a schematic smooth interpolation with the derived asymptotes, not a precision transfer-function fit. The linear ideal-fluid model excludes baryonic acoustic structure and small-scale nonlinear evolution.

There is also a gauge and horizon qualification. For a mode still outside the Hubble radius at $z=1$, the printed Newtonian-gauge density has ${\delta }_c\simeq -2\Phi$, and its formal dimensional spectrum is instead proportional to $k^{-3}$. The conventional large-scale $k$ branch describes modes that are large relative to the equality scale but already subhorizon at the observation time. Alternatively, the comoving matter density, ${\Delta }_c={\delta }_c+3\mathcal{H}{\theta }_c/k^2$ with ${\theta }_c=\nabla \cdot {\mathbf{v}}_c$, removes that constant gauge term and obeys ${\Delta }_c=-2k^2\Phi /(3{\mathcal{H}}^2)$ in the growing matter solution. The usual matter-spectrum sketch can be continued to small $k$ in this comoving-density convention. The dimensional spectrum requested here is $P$, rather than the dimensionless cosmological power spectrum $k^{3}P/(2{\pi }^2)$, whose slopes would differ by three.

Use units $c=\hslash =k_B=1$, and write $H=\dot{a}/a$ for the Hubble parameter, with dots denoting cosmic time. Differentiating the Friedmann equation givesOn an interval where $H\ne 0$, use the Hubble parameter identity $\dot{H}=\ddot{a}/a-H^2$ and the Friedmann acceleration equation to findSubstitution cancels the curvature term and gives the continuity equationThe cosmological perfect-fluid continuity equation extends by continuity through a regular isolated turning point. Physically it states that the change of energy in a comoving volume is the negative of the pressure work: $d(\rho a^3)=-Pd(a^3)$.

For separately conserved cosmological fluids, each component obeys this equation individually. A constant equation-of-state parameter $w_i=P_i/{\rho }_i$ therefore giveswhere the present scale factor is normalized to $a_0=1$. Thus the component density laws areThe extra factor for radiation in cosmology is the cosmological redshift of each photon's energy; pressureless matter has only number dilution, while the specified dark energy is a cosmological constant.

The critical density at a given expansion rate is the total density that makes the spatial curvature vanish:These cosmological density parameters use the critical density at that same time. Put $E(a)=H(a)/H_0$ and ${\Omega }_{K,0}=-K/H_0^2$. The Friedmann equation yieldsConsequently the fractional-density evolution, rather than just the component-density evolution, isIn particular ${\sum }_i{\Omega }_i=1+K/(a^2{H}^2)$. The simple powers of $a$ alone apply to ${\rho }_i$, or to ${\rho }_i/{\rho }_{\mathrm{crit},0}$, not to the instantaneous cosmological density parameters. At a recollapse turning point $H=0$, those instantaneous ratios are undefined even though the component densities remain finite.

In the local orthonormal tetrad, the components of the photon number current areThey are respectively the photon number density and the photon number flux. Thus $J^{\mu }$ is the photon number four-current, not an energy current or an electric charge current. Photons carry no electric charge. For the isotropic background,Integrating the covariant Boltzmann equation over the invariant mass-shell measure, with vanishing momentum-boundary flux, giveswhere $C[f]$ is the collision operator in the covariant streaming equation. ThereforeCollisionless photon propagation satisfies this condition, as do Thomson scattering and ordinary Compton scattering when all participating photons are counted: each collision has one incoming and one outgoing photon. Photon emission, photon absorption, bremsstrahlung, double Compton scattering and electron-positron annihilation in cosmology or its inverse generally give a nonzero number source. A cancellation of net creation and destruction can nevertheless make the integrated source vanish even when individual number-changing reactions occur. Cosmological redshift itself changes photon energy, not photon number; in an expanding homogeneous universe a conserved photon number current gives ${\bar{n}}_{\gamma }\propto a^{-3}$.

For comoving galaxies their relative comoving vector $\mathbf{\chi }$ is constant, and the physical displacement is $\mathbf{r}(t)=a(t)\mathbf{\chi }$, where $a$ is the scale factor. Differentiating gives the Hubble lawPeculiar velocity relative to this comoving frame is excluded in this relation.

Transport a wave crest between neighbouring comoving observers separated by a small physical distance $d\ell$. Their recession speed is $Hd\ell$, while the light travel time is $dt=d\ell /c$ to first order. The local Doppler shift therefore changes its wavelength by $d\lambda /\lambda =Hdt$. Multiplying these infinitesimal shifts along the ray, or integrating their logarithms, yieldsThe use of the local first-order Doppler law is exact in the limit of infinitesimal steps: the accumulated second-order errors tend to zero. A single special-relativistic recession velocity assigned to widely separated galaxies would not justify this cosmological redshift relation.

For a component with constant barotropic equation of state $P=w\rho$, the cosmological perfect-fluid continuity equation givesHence ${\rho }_i={\rho }_{i,0}{a}^{-3(1+w_i)}$. Choose the present scale factor to be $a_0=1$, so the cosmological redshift obeys $1+z=a^{-1}$, and define the present cosmological density parameter bySubstitution in the spatially flat Friedmann equation then yields

The Soltan argument compares the time-integrated luminosity density of the cosmological AGN population with the present comoving mass density in dormant supermassive black holes. If $U_{\mathrm{AGN}}$ is the emitted energy density corrected for obscuration and bolometric output, accretion with population-averaged efficiency $\bar{\eta }$ predictsIn practice $U_{\mathrm{AGN}}$ comes from integrating AGN luminosity functions over luminosity and cosmological redshift, with corrections for obscured sources and missed wavebands, while ${\rho }_{\mathrm{BH}}$ is inferred from local galaxy--black-hole scaling relations.

The inferred efficiency is of order the canonical thin-disk value, about ten per cent, so most cosmic black-hole mass was accumulated in radiatively efficient, optically thick accretion episodes. Radiatively inefficient flows can dominate low-luminosity activity or brief extreme phases, and mergers redistribute existing mass, but neither naturally accounts for the observed integrated AGN radiation while supplying most of the final mass.

The given formula is the integral of the Planck photon distribution. SetThenwhereEach photon of frequency $\nu$ has energy $h\nu$, so the energy density isThis is the temperature scaling of thermal photon number and energy densities.

After decoupling, cosmological redshift givesThe occupation number is conserved along the freely propagating photon trajectory. Its exponential argument at decoupling becomesThus the spectrum still has the Planck form ifFrequency and temperature redshift by the same factor, which is the redshift preservation of a thermal photon spectrum.

A suitable Spatially flat FLRW metric isBoth source and detector are comoving, so cosmological redshift stretches the wave period in proportion to the scale factor. If $f_e$ is the local source frequency at emission and $f_o$ is the frequency measured in Earth's local frame, thenThe same relation applies to gravitational waves propagating on the FLRW background in the geometric-optics regime.

For successive radial null rays from a comoving source, ${\int }_{t_1}^{t_0}cdt/R(t)$ equals the same spatial integral. Subtracting two nearby wavecrests gives $\delta t_0/R_0=\delta t_1/R_1$, hence the cosmological redshiftFor transparent propagation, photon energies are reduced by $1+z$ and arrival intervals increased by $1+z$. The observer-area sphere is $4\pi (R_{0}r{)}^2$, so the observed bolometric flux isThus the requested $D_L=(1+z)D$ uses $D=R_{0}r$, the present transverse proper/area distance. In a curved metric this is not the radial proper distance $R_0{\int }_0^{r}d{r}^{\prime }/\sqrt{1-K{r}^{\prime 2}}$; the two coincide for $K=0$. The source's unqualified “proper distance” needs this distinction.

Type Ia supernovae are standardizable candles: light-curve shape and colour calibrate their luminosity. Distant events are fainter, hence have larger inferred luminosity distances, than appropriate matter-only decelerating models predict. Their redshift–distance relation favours recent accelerated expansion. In general relativity, sufficiently negative pressure, $\rho +3P/c^2<0$, drives acceleration. Dust, evolution, calibration and selection are checked as possible systematics; a single faint supernova is not the argument.^[ref]

An independent complementary test combines the cosmic microwave background acoustic distance/near-flat geometry with matter-density and baryon acoustic oscillation constraints. Matter alone does not supply the required late-time expansion history; a dark-energy component fits the joint data. CMB geometry alone has curvature/expansion degeneracies and is not an assumption-free detection of a cosmological constant.^[ref]

In natural units $c=\hslash =1$, a regulated zero-point energy estimate isThe integral is ultraviolet divergent and requires renormalization; it is not a finite prediction from counting “virtual pairs.” An electroweak scale estimate gives $|{\rho }_{\mathrm{vac}}|\sim m_t^4=(174{)}^4\mathrm{Ge}{\mathrm{V}}^4\simeq 9.2\times 10^8\mathrm{Ge}{\mathrm{V}}^4$, or somewhat smaller with loop factors. Compared with $10^{-47}\mathrm{Ge}{\mathrm{V}}^4$ this is roughly $10^{56}$ too large.

The source's units are inconsistent: $10^{-47}\mathrm{Ge}{\mathrm{V}}^4$ is a vacuum energy density, whereas $8\pi G{\rho }_{\mathrm{vac}}$ has units $\mathrm{Ge}{\mathrm{V}}^2$. The comparison above consistently uses densities. Bare and quantum contributions must cancel to extraordinary accuracy if this naive scale estimate is representative: the cosmological constant problem. Its severity is not a measured value of an individual cutoff integral.