ΛCDM Has Failed

The Friedmann-Lemaître-Robertson-Walker (FLRW) metric forms the foundation of modern cosmology ΛCDM, but its adoption stems from mathematical convenience rather than physical accuracy or predictive power.

The metric oversimplifies Einstein's field equations with idealized assumptions of perfect homogeneity and isotropy. These assumptions contradict observable structures at all scales. The universe is not homogeneous on any scale. Mass density is a scale-dependent variable.

The homogeneous assumption was never intended as a literal representation of the universe's structure. It was only a tool to enable a mathematically tractable model of cosmic dynamics.

And while there have been many attempts to reconcile the smaller scales by using the FLRW as a background, they all fail to capture the full geometry of nested gravitational wells.

A New Framework

This model proposes a fundamentally different approach:

Step 1: Interpet the Hubble constant as gravitational redshift.

Step 2: Derive the distribution from the Hubble constant.

Step 3: Formulate and stress test the resulting model.

All matter in the universe is connected to a vast gravitational network, from subatomic particles to the observable universe. What we see as individual "objects" are merely convenient mental forms created by our brains.

In reality, these boundaries are all arbitrary human impositions on a seamless manifold.

Building the Metric

Consider the path a photon takes as it is climbing out of a series of nested gravitational wells. These paths can be used as the reference lines for our coordinates. Null geodesics are a great choice because they naturally encode the causal structure of spacetime on all scales without relying on any particular preferred frame of reference.

The line element is written as:

$$ ds^2 = -\left(1 - \frac{Hr}{c}\right)^2dt^2 + \frac{dr^2}{\left(1 - \frac{Hr}{c}\right)^2} + r^2d\Omega^2 $$

This metric produces null geodesics that follow:

$$ \frac{dt}{dr} = \pm\frac{1}{\left(1 - \frac{Hr}{c}\right)^2} $$

These equations establish the basic geometric structure without requiring additional cosmological assumptions.

Of course we should expect local fluctuations, but as the scale increases so does the symmetry of the overall structure.

Matter Distribution

The universe’s mass is nested like Matryoshka dolls. The density follows a linear drop up to the horizon
$$ r_s = \frac{c}{H}. $$

Einstein’s field equations lead to
$$ \rho(r) \;=\; \frac{3H^2}{8\pi G}\Bigl(1 - \frac{H\,r}{c}\Bigr), \quad 0 \le r < r_s, $$
where \(H\) is the Hubble‐like parameter, \(G\) is the gravitational constant, and \(c\) is the speed of light. As \(r\) approaches \(r_s\), the term \( \Bigl(1 - \frac{H\,r}{c}\Bigr) \) goes to zero, bringing the density \(\rho\) to zero and fixing the total mass.

This hierarchical distribution prevents mass from collapsing to a small region; instead, mass extends nearly out to \(r_s\). The enclosed mass,
$$ M(r) = 4\pi \int_{0}^{r} \rho(r')\,r'^{2} \, dr', $$
accumulates significantly over a broad range of radii before tapering off near the horizon. In this way, the potential well increases along \(r\).

It is quite remarkable how well this distribution aligns with observations, especially when you consider the FLRW assumes a homogeneous distribution.

Flat Rotational Curves

With mass extending across large radii, the enclosed mass \(M(r)\) does not saturate quickly. Consequently, orbital velocities stay higher than expected from a simple "Keplerian" falloff. This prolonged buildup of mass ensures that \(v(r)\) remains nearly constant—or "flat"—over a wide radial range.

$$ v(r) = \sqrt{\frac{G \, M(r)}{r}}, $$

Near the horizon \(r_s\), the density goes to zero, eventually reducing further mass accumulation. Thus, rotation speeds can drop only when approaching \(r_s\). In essence, this linear density profile naturally gives rise to extended gravitational potentials that sustain flat rotation curves well beyond central regions.

$$ \rho(r) \;\propto\; \Bigl(1 - \frac{H\,r}{c}\Bigr), $$

In observational terms, such behavior parallels the flat rotation profiles seen in galaxies, where velocities do not drop off as \(r^{-1/2}\). Here, it emerges from a horizon-based spacetime structure, showing how cosmic-scale parameters (\(H\), \(r_s\)) can govern local rotational dynamics in a self-consistent way.

The Hubble Tension

The “Hubble tension” refers to a persistent discrepancy between local and early-universe measurements of the Hubble constant \(\displaystyle H_0\).

Local (distance-ladder) observations often find \(\displaystyle H_0 \approx 73\text{–}74\ \mathrm{km\,s^{-1}\,Mpc^{-1}}\), while cosmic microwave background data typically yield \(\displaystyle H_0 \approx 67\text{–}68\ \mathrm{km\,s^{-1}\,Mpc^{-1}}\).

This mismatch, significant at around the 5\(\,\sigma\) level, suggests potential issues with systematics or the need for new physics beyond the standard model of cosmology.

Researchers explore improved calibration methods, alternative distance indicators, and modified cosmological frameworks to resolve the tension.

Thus, \(\displaystyle H_0\) remains one of the most debated parameters in modern cosmology.

This tension will always remain within \(\displaystyle \Lambda\mathrm{CDM}\) because of its interpretation of the redshift. In reality, local and cosmic redshift measurements arise from the same gravitational potential, defined by \(\displaystyle H\) and the horizon \(\displaystyle r_{s} = \frac{c}{H}\).

For small distances (\(\displaystyle r \ll r_s\)), the redshift \(\displaystyle z \approx \frac{H\,r}{c}\) naturally matches the local “distance-ladder” Hubble constant of \(\displaystyle 73\text{–}74\ \mathrm{km\,s^{-1}\,Mpc^{-1}}\).

On large (cosmic) scales, integrating over the entire potential well yields an effective rate aligning with measurements at extreme distances, \(\displaystyle 67\text{–}68\ \mathrm{km\,s^{-1}\,Mpc^{-1}}\), thus removing any conflict.

Because both local and global measurements are tied to the same underlying geometry, the discrepancy commonly seen as the “Hubble tension” is not an anomaly.

The Observable Limit

The Schwarzschild radius marks a boundary in spacetime where gravity becomes so strong that nothing can escape. For a spherically symmetric mass \(\displaystyle M\), this occurs at radius \(\displaystyle r_s = \frac{2GM}{c^2}\). At this radius, the Schwarzschild metric exhibits a coordinate singularity where the time component vanishes and the radial component becomes infinite: \(\displaystyle g_{tt} = -\Bigl(1 - \frac{2GM}{r\,c^2}\Bigr)\) and \(\displaystyle g_{rr} = \Bigl(1 - \frac{2GM}{r\,c^2}\Bigr)^{-1}\).

Our model predicts a horizon at \(\displaystyle r_s = \frac{c}{H}\) with geometric similarities to a Schwarzschild black hole. Both metric components scale with \(\displaystyle \Bigl(1 - \frac{H\,r}{c}\Bigr)^2\) rather than being reciprocals, and the density profile \(\displaystyle \rho(r) = \frac{3H^2}{8\pi G}\Bigl(1 - \frac{H\,r}{c}\Bigr)\) smoothly approaches zero at the horizon. This cosmological horizon is a coordinate singularity that only arises because of the extreme nature of the chosen coordinates.

CMB as Hawking Radiation

While the predicted horizon \(r_s\) is fundamentally different than the event horizon of a black hole, the gravitational potential is still extreme at \(r_s\).

The Hawking temperature of the horizon is given by:

$$ T_H = \frac{\hbar c^3}{8\pi GM_sk_B} $$

The equation can also be formulated with \(r_s\).

$$T(r_s) \;=\; \frac{\hbar \, c}{4 \,\pi \, r_s \, k_B}$$

As \(r_s\) increases, the temperature of emitted radiation approaches zero.

However, unlike the radiation escaping a traditional black hole, the radiation in this model undergoes an extreme blue shift because the observer is a distance of \(r_s\) from the location of emission.

If we interpret the CMB as this Hawking radiation, we would need a blue shift factor of around \(1.9\times 10^{30}\).

$$\frac{2.725\,\mathrm{K}}{1.4\times10^{-30}\,\mathrm{K}}\;\approx\;1.9\times 10^{30}$$.

While this sounds like an absurd factor of change, it represents the scales allowable by relativity.

Resolving Anomalies in the CMB

There are a number of unresolved issues within the mainstream interpretation of the Cosmic Microwave Background. These anomalies can be resolved within a static framework.

Quadrupole--Octopole Alignment - Axis of Evil Anomaly

A well-known anomaly is the apparent alignment of the CMB quadrupole and octopole. With our static metric, a small off-center displacement \(\Delta r_0\) of the observer from \(r=0\) modifies the gravitational redshift factor

$$ z(r) \;\approx\; \bigl[\,1 - \tfrac{H\,(r+\Delta r_0)}{c}\bigr]^{-1} \;-\; 1.$$

When extended to near-horizon scales, any large-scale density or potential gradient can couple to these lowest multipoles (i.e.\ \(\ell=2,3\)) and introduce a preferred axis. Thus, an observer not exactly at the metric center naturally sees a preferred direction set by \(\Delta r_0\), aligning the low-\(\ell\) modes.

Hemispherical Power Asymmetry

CMB maps suggest there is a modest but persistent asymmetry in fluctuation power between opposite hemispheres. In this framework, such an asymmetry emerges if \(\Delta r_0 \neq 0\) implies a slightly different net redshift on one side of the sky versus the other.

Quantitatively, the fractional temperature difference can scale as

$$ \frac{\Delta T}{T} \;\sim\; \frac{H\,\Delta r_0}{c} $$

since the gravitational redshift factor \((1 - Hr/c)\) is not isotropic when viewed from an off-center vantage. This small anisotropic shift in the effective temperature can feed directly into an observed hemispherical power contrast.

CMB Anomalies cont.

Low Quadrupole and Large-Scale Power Deficit

The suppressed amplitude of the quadrupole (and other large angular scales) is another puzzle in standard FRW cosmology. In the static model, boundary conditions at the universal horizon \(r_s = c/H\) and the vanishing density \(\rho(r_s)=0\) can effectively damp large-scale modes.

\[ T_{\rm CMB} \;\simeq\; \bigl(1 - \tfrac{Hr}{c}\bigr)^{-1}\,T_H, \]

so that modes most sensitive to the near-horizon region (where \(\bigl(1 - \tfrac{Hr}{c}\bigr)\to 0\)) can experience additional suppression or phase alignment. This mechanism can imprint a deficit in the lowest multipoles relative to naive scale-invariant expectations.

The Cold Spot

A prominent large-angle CMB feature is the "cold spot," often attributed to a supervoid. In the static scenario, an extra underdensity along a line of sight effectively increases the path-integrated redshift. A local deficit \(\Delta\rho(r) < 0\) yields additional potential depth, causing photons crossing that region to lose more energy. A rough estimate of the temperature decrement can be obtained via

$$\Delta T_{\rm cold} \;\sim\; \int \!\Delta\Phi(r)\,dr,$$

where \(\Delta\Phi(r)\) is the locally perturbed gravitational potential. Even a modest underdensity becomes magnified when viewed against the full horizon-scale backdrop, producing a distinctly colder patch in the CMB.

Derivation of the Planck Length

The relationship between the observed CMB temperature (T _CMB = 2.725 K) and the Hawking temperature is given by the maximum relativistic blueshift factor:

$$ \frac{T_\text{CMB}}{T_H} \;=\; \sqrt{\frac{r_s}{2\,l_p}} \;=\; \sqrt{\frac{c}{2\,H\,l_p}} \;=\; 1.995090 \times 10^{30} $$

This shows that the CMB can be interpreted as maximally blueshifted Hawking radiation from the universal horizon, with the blueshift determined by the ratio of the horizon scale to the Planck scale.

The relationship between the CMB temperature and Hawking radiation from the universal horizon provides a remarkable bridge between quantum and classical gravity. Of particular significance is the derivation of the Planck length through observable parameters. One finds that

$$ l_p = \sqrt{\frac{\hbar G}{c^3}} = \sqrt{\frac{\hbar}{c^3}\;\cdot\;\frac{3H}{8\pi\rho}} = \sqrt{\frac{3\hbar H}{8c^2\pi\rho}} $$

Quantum Gravity

QFT demands discrete mass-energy for fundamental particles and excitations, meaning that mass is inherently quantized. Since general relativity links mass to spacetime curvature via Einstein’s field equations, the discrete nature of mass implies a corresponding discretization of spacetime: no separate mechanism is needed to “quantize gravity.”

The ultraviolet (UV) cutoff arises at the Planck length,

\[ l_p \;=\; \sqrt{\frac{\hbar\,G}{c^3}}, \]

while the infrared (IR) cutoff appears at the horizon,

\[ r_s \;=\; \frac{c}{H}. \]

With these natural cutoffs, the vacuum energy density remains near observed levels, rather than blowing up as naive Planck-scale estimates would suggest. For instance,

\[ \rho_{\text{vac}} \;=\; \frac{\pi\,\hbar\,c}{4\,l_{\text{UV}}^2\,l_{\text{IR}}} \]

is significantly smaller than the usual predictions. Light traveling on null geodesics “samples” this geometry, so large-scale gravitational phenomena can encode quantum effects without requiring an additional quantization procedure for spacetime itself.

Moreover, the effective gravitational coupling gains a mild radial dependence,

\[ G_{\text{eff}}(r) \;=\; G\Bigl(1 + \frac{G\,\hbar}{2\pi\,c^3\,r^2}\Bigr), \]

but stays perturbative at all scales. By tying the Planck length to the cosmic horizon, this setup bridges the smallest and largest scales, linking discrete mass sources to a quantized curvature background in a direct, testable manner.

Review

The model we have presented so far can be considered superior to \(\displaystyle \Lambda\mathrm{CDM}\) in many ways.

• Constructed from General Relativity alone.
• Interprets the Hubble constant as a result of gravitational redshift.
• Interprets the CMB as a result of blue shifted Hawking radiation.
• Results in a stable, static, and valid distribution of mass.
• No need for expansion.
• No need for Dark Energy.
• No need for a cosmological constant.
• No need for inflation.
• No need for Dark Matter
• Resolves the Huble tension
• Resolves multiple anomalies in the CMB
• Capable of deriving the Planck length.