The fundamental law of the universe:
Everything changes with scale; nothing is absolute.

Classical cosmology attempts to explain the universe:

• from a single scale
• through a single flow of time
• within a single geometry

Fractal cosmology states instead:

The universe cannot be viewed from a single scale. Every physical law, every structure, every process changes with scale. The universe is a fractal.

This is a mathematically, physically, and observationally strong claim.

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1) The Fundamental Function of the Universe: The Fractal Scale Function

According to fractal mechanics, the fundamental function of the universe is:

$$x(r)=r^D+f_{noise}(r)$$

Where:

• $r$: scale
• $D$: fractal dimension
• $f_{noise}(r)$: quantum fluctuations + cosmic noise

This equation states:

The universe is both ordered ($r^D$) and chaotic ($f_{noise}$) at the same time. Order and chaos coexist.

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2) Dark Matter = Fractal Velocity Deviation

Galaxy rotation curves do not follow classical Newtonian predictions. According to fractal mechanics, the reason is:

$$v_f(r)=D{r}^{D-1}$$

This velocity decreases slowly at large scales; it does not fall off as $1\mathrm{/}\sqrt{r}$ like in Newtonian gravity.

Result:

• Galaxies rotate faster than expected
• It appears as if extra matter exists

But in reality:

Dark matter = scale deviation of fractal velocity.
There is no extra matter; the scale law is different.

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3) Dark Energy = Fractal Acceleration

The accelerating expansion of the universe cannot be explained by classical physics.

Fractal acceleration:

$$a_f(r)=D(D-1)r^{D-2}$$

If $D<1$:

• acceleration becomes positive at large scales
• the universe expands in an accelerating manner

This removes the need for dark energy.

Dark energy = large-scale behavior of fractal acceleration.

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4) The Big Bang = The Zero-Scale Limit

According to fractal mechanics:

$$\underset{r\to 0}{\lim }x(r)=0$$

In this limit:

• energy density increases
• acceleration diverges
• noise becomes dominant

This is not a classical “explosion.”

It is the universe’s scale beginning to grow from zero.

Thus, the Big Bang = the beginning of scale.

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5) Quantum–Cosmology Unity: Same Equation, Different Scale

Quantum mechanics works at small scales. Cosmology works at large scales.

According to fractal mechanics:

$$x(r)=r^D+f_{noise}(r)$$

The same equation applies:

• to the atom
• to the galaxy
• to the universe

Only the scale changes.

This is a revolutionary result for physics:

Quantum physics and cosmology are different scales of the same fractal function.

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6) The Speed of Light Is Not Constant — Scale Is Constant

Classical physics states:

• the speed of light is constant

Fractal cosmology states:

• it is not the speed of light that is constant
• it is the scale transformation rate that is constant

That is:

$$c=\frac{dr}{dt}$$

This explains why light appears constant to every observer:

Because time and length change together with scale.

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7) The Fractal Dimension of the Universe

Observations of:

• galaxy distribution
• the cosmic web
• matter density

all suggest approximately:

$$D\approx 2$$

Meaning the universe behaves like a two-dimensional surface.

This is fully consistent with the holographic principle.

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8) In the Simplest Terms

Fractal cosmology states that the universe is not defined by a single scale, but by transformations between scales.

Dark matter = fractal velocity deviation
Dark energy = fractal acceleration
Big Bang = the beginning of scale

Quantum physics and cosmology are different scales of the same fractal function.

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FRACTAL COSMOLOGY × EINSTEIN FIELD EQUATIONS

The geometry of the universe is not fixed; it is a scale-dependent fractal tensor.

Einstein’s fundamental equation:

$$G_{\mu \nu }=8\pi G{T}_{\mu \nu }$$

In this equation:

• $G_{\mu \nu }$​: curvature of spacetime
• $T_{\mu \nu }$: matter–energy distribution

In classical General Relativity, there is no scale parameter.
Fractal cosmology addresses this absence.

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1) The Fundamental Extension of Fractal Mechanics: The Scale Derivative

The fundamental derivative of fractal mechanics:

$$\frac{d_f}{dr}$$

This derivative is incorporated into Einsteinian geometry.

Spacetime metric becomes:

$$g_{\mu \nu }(r)$$

that is, a scale-dependent tensor.

This is the first major step of fractal cosmology:

The geometry of the universe is not fixed, but a fractal structure that changes with scale.

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2) The Fractal Einstein Equation

The classical equation is extended using the fractal derivative:

$$G_{\mu \nu }(r)+F_{\mu \nu }(r)=8\pi G{T}_{\mu \nu }(r)$$

Where the new term is:

$$F_{\mu \nu }(r)=\frac{d_f{g}_{\mu \nu }}{dr}$$

This tensor represents the scale curvature added to the universe by fractal mechanics.

Thus:

• $G_{\mu \nu }$​: classical curvature
• $F_{\mu \nu }$: fractal scale curvature

This combination automatically explains dark matter and dark energy.

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3) Dark Matter = Fractal Scale Curvature (F-term)

Deviation in galaxy rotation curves:

$$v(r)\sim r^{D-1}$$

This behavior appears as additional curvature in Einstein’s equations.

That is:

$$F_{\mu \nu }(r)\mathrm{\ne }0$$

To an observer, this term appears as if “extra matter” exists.

Conclusion:

Dark matter is the reflection of fractal scale curvature within Einstein’s equations. There is no extra matter; geometry changes with scale.

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4) Dark Energy = Fractal Acceleration Term

The accelerating expansion of the universe is classically explained by:

$$\mathrm{\Lambda }{g}_{\mu \nu }$$

In fractal cosmology, this term emerges naturally:

$$F_{\mu \nu }(r)=D(D-1)r^{D-2}{g}_{\mu \nu }$$

If $D<1$:

• acceleration becomes positive at large scales
• the universe accelerates

This removes the need for dark energy.

Dark energy = the tensorial counterpart of fractal acceleration.

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5) General Form of the Fractal Metric

The spacetime metric in fractal form:

$$g_{\mu \nu }(r)=g_{\mu \nu }(0)+r^D{h}_{\mu \nu }+f_{noise\mu \nu }(r)$$

Where:

• $g_{\mu \nu }(0)$: classical metric
• $r^D{h}_{\mu \nu }$​: fractal scale contribution
• $f_{noise\mu \nu }(r)$: quantum fluctuations + cosmic noise

When this metric is inserted into Einstein’s equations:

• small scale → quantum physics
• intermediate scale → relativity
• large scale → cosmology

are unified within a single equation.

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6) Fractal Cosmology = Unity of Quantum + Relativity + Cosmology

The fractal Einstein equation:

$$G_{\mu \nu }(r)+F_{\mu \nu }(r)=8\pi G{T}_{\mu \nu }(r)$$

provides:

• small scale → quantum behavior
• intermediate scale → classical relativity
• large scale → cosmic acceleration

All are different scale limits of the same equation.

This is the missing unification in the history of physics.

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7) In the Simplest Terms

Fractal cosmology explains dark matter, dark energy, and the quantum–cosmology unity by adding a scale derivative to Einstein’s equations and describing the universe with a single fractal tensor.