Cosmological Redshift: Cosmic Shear of Time

“Redshift can be understood as the stretching of phase rhythm by the cosmic scale factor.”

🎯 Core Proposition

Theoretical Proposition (Redshift as Phase Rhythm Ratio):

In FRW cosmology, cosmological redshift can be expressed within the GLS framework as:

$$\boxed{1+z=\frac{a(t_0)}{a(t_e)}=\frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}}$$

where:

• $a(t)$: cosmic scale factor
• $t_0$: observation time
• $t_e$: emission time
• $\varphi$: photon eikonal phase

Physical Interpretation:

• Left side: observed redshift
• Middle: scale factor ratio (standard formula)
• Right side: phase “rhythm” ratio (GLS formulation)
• Inference: Redshift can be viewed as a global rescaling of the time scale.

graph TB
    EM["Emission<br/>t_e, a(t_e)"] --> PHOTON["Photon Propagation<br/>Null Geodesic"]
    PHOTON --> OBS["Observation<br/>t_0, a(t_0)"]

    EM --> PE["Emission Frequency<br/>ν_e = dφ_e/dt"]
    OBS --> PO["Observation Frequency<br/>ν_0 = dφ_0/dt"]

    PE --> Z["Redshift<br/>1+z = ν_e/ν_0"]
    PO --> Z
    Z --> A["Scale Factor<br/>= a₀/a_e"]

    style EM fill:#e1f5ff
    style PHOTON fill:#ffe1e1
    style Z fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style A fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

💡 Intuitive Image: Expanding Rubber Band of the Universe

Analogy: Stretched Wave

Imagine drawing a sine wave on a rubber band:

Original: ∿∿∿∿∿∿∿  (wavelength λ_e)
Stretched: ∿  ∿  ∿  (wavelength λ_0 = (1+z)λ_e)

Stretching Process:

• Rubber band length $L(t)=a(t)L_0$
• Wavelength stretches accordingly $\lambda (t)=a(t){\lambda }_0$
• Frequency decreases $\nu (t)=c/\lambda (t)\propto 1/a(t)$

Redshift:

$$1+z=\frac{{\lambda }_0}{{\lambda }_e}=\frac{a(t_0)}{a(t_e)}$$

Physical Meaning: Redshift measures the “stretching factor” of cosmic expansion.

Slowing of Clocks

Another Perspective: The “clock” at the emission end slows down when observed

Atomic Transition:

• Emission: frequency ${\nu }_e$
• Observation: frequency ${\nu }_0={\nu }_e/(1+z)$

Phase Accumulation Rate:

Emission: $d{\varphi }_e/dt=2\pi {\nu }_e$

Observation: $d{\varphi }_0/dt=2\pi {\nu }_0=2\pi {\nu }_e/(1+z)$

Ratio:

$$\frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}=1+z$$

Physical Interpretation: Redshift manifests as a global scaling of “phase rhythm”.

graph LR
    EMIT["Emission End<br/>ν_e, dφ_e/dt"] --> SPACE["Cosmic Expansion<br/>a(t) increases"]
    SPACE --> OBS["Observation End<br/>ν_0 = ν_e/(1+z)"]
    OBS --> SLOW["Phase Slows<br/>dφ_0/dt < dφ_e/dt"]

    style EMIT fill:#e1f5ff
    style SPACE fill:#ffe1e1
    style SLOW fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

📐 FRW Cosmology

FRW Metric

Metric:

$$d{s}^2=-d{t}^2+a(t{)}^2{\gamma }_{ij}d{x}^{i}d{x}^j$$

where ${\gamma }_{ij}$ is the constant curvature 3-dimensional spatial metric:

• $k=+1$: 3-sphere (closed universe)
• $k=0$: flat (flat universe)
• $k=-1$: hyperbolic (open universe)

Friedmann Equation:

$${\left(\frac{\dot{a}}{a}\right)}^2=\frac{8\pi G}{3}\rho -\frac{k}{a^2}+\frac{\Lambda }{3}$$

Acceleration Equation:

$$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho +3p)+\frac{\Lambda }{3}$$

Comoving Observer

Definition: Observer moving with cosmic expansion, coordinates $(t,\mathbf{x})$ fixed.

4-Velocity:

$$u^{\mu }=(1,0,0,0)$$

Proper Time:

$$d\tau =dt$$

Physical Meaning: Comoving observer’s clock measures cosmic time $t$.

Photon Null Geodesics

Null Geodesic Equation: $d{s}^2=0$

$$-d{t}^2+a(t{)}^2{\gamma }_{ij}d{x}^{i}d{x}^j=0$$

Radial Propagation ($d\theta =d\varphi =0$):

$$\frac{dt}{a(t)}=\pm d\chi$$

where $\chi$ is the comoving radial coordinate.

Conformal Time:

$$d\eta =\frac{dt}{a(t)}$$

Null Geodesics: $d\eta =\pm d\chi$ (straight lines).

graph TB
    FRW["FRW Metric<br/>ds² = -dt² + a²dΣ²"] --> COMOVING["Comoving Observer<br/>u = ∂_t, dτ = dt"]
    FRW --> NULL["Null Geodesics<br/>ds² = 0"]
    NULL --> ETA["Conformal Time<br/>dη = dt/a"]
    ETA --> STRAIGHT["Straight Lines<br/>dη = ±dχ"]

    style FRW fill:#e1f5ff
    style ETA fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style STRAIGHT fill:#e1ffe1

🌀 Redshift Formula Derivation

Standard Derivation

Photon 4-Momentum:

$$k^{\mu }=\frac{d{x}^{\mu }}{d\lambda }$$

where $\lambda$ is the affine parameter.

Null Geodesic: $k_{\mu }{k}^{\mu }=0$

Along Geodesic: $k^{\nu }{\nabla }_{\nu }{k}^{\mu }=0$

For a comoving observer, define frequency:

$$\nu =-k_{\mu }{u}^{\mu }=k_0=\frac{dt}{d\lambda }$$

Derivative:

$$\frac{d\nu }{d\lambda }=k^{\alpha }{\nabla }_{\alpha }{k}_0$$

Using Christoffel symbols (omitted), we get:

$$\frac{d\nu }{\nu }=-\frac{da}{a}$$

Integrating:

$$\ln \frac{{\nu }_0}{{\nu }_e}=-\ln \frac{a(t_0)}{a(t_e)}$$

Therefore:

$$\boxed{\frac{{\nu }_0}{{\nu }_e}=\frac{a(t_e)}{a(t_0)}}$$

Redshift Definition:

$$1+z:=\frac{{\nu }_e}{{\nu }_0}=\frac{a(t_0)}{a(t_e)}$$

Phase Rhythm Derivation

Eikonal Approximation:

Photon wavefunction $\psi \sim e^{i\varphi /\hslash }$, where $\varphi$ is the phase.

4-Momentum:

$$k_{\mu }={\partial }_{\mu }\varphi$$

Frequency:

$$\nu =-\frac{1}{2\pi }{k}_{\mu }{u}^{\mu }=\frac{1}{2\pi }\frac{\partial \varphi }{\partial t}$$

For a comoving observer:

$$\nu =\frac{1}{2\pi }\frac{d\varphi }{dt}$$

From $\nu \propto 1/a$:

$$\frac{d\varphi }{dt}\propto \frac{1}{a(t)}$$

Emission and Observation:

$$\frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}=\frac{a(t_0)}{a(t_e)}=1+z$$

Conclusion: Phase rhythm ratio is numerically equal to redshift.

graph TB
    PHI["Phase<br/>φ(t, x)"] --> K["4-Momentum<br/>k_μ = ∂_μφ"]
    K --> NU["Frequency<br/>ν = -k·u/2π"]
    NU --> PROP["Propagation<br/>ν ∝ 1/a(t)"]
    PROP --> RATIO["Rhythm Ratio<br/>(dφ/dt)_e / (dφ/dt)_0"]
    RATIO --> Z["Redshift<br/>1+z = a₀/a_e"]

    style PHI fill:#e1f5ff
    style RATIO fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style Z fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🔑 Redshift as Time Rescaling

Rescaling of Cosmic Time

Local Time Scale:

At emission point $(t_e,{\mathbf{x}}_e)$:

$${\kappa }_e=\frac{1}{2\pi }\frac{d\varphi }{dt}{|}_e$$

At observation point $(t_0,{\mathbf{x}}_0)$:

$${\kappa }_0=\frac{1}{2\pi }\frac{d\varphi }{dt}{|}_0$$

Ratio:

$$\frac{{\kappa }_e}{{\kappa }_0}=1+z$$

Physical Interpretation:

Redshift can be viewed as a global rescaling factor of the local time scale.

Time Dilation:

For the same physical process (e.g., supernova explosion), observed duration:

$$\Delta t_{\text{obs}}=(1+z)\Delta t_{\text{proper}}$$

This is exactly “cosmological time dilation”.

Distance-Redshift Relation

Luminosity Distance:

$$d_L=(1+z){\int }_0^z\frac{cd{z}^{\prime }}{H(z^{\prime })}$$

where $H(z)=H_0\sqrt{{\Omega }_m(1+z{)}^3+{\Omega }_{\Lambda }}$.

Angular Diameter Distance:

$$d_A=\frac{d_L}{(1+z{)}^2}$$

Hubble’s Law (low redshift):

$$z\approx \frac{H_{0}d}{c}$$

Physical Meaning: By measuring redshift, we can infer distance and cosmic evolution.

graph LR
    Z["Redshift<br/>z"] --> DL["Luminosity Distance<br/>d_L ∝ ∫dz/H(z)"]
    Z --> DA["Angular Diameter Distance<br/>d_A = d_L/(1+z)²"]
    Z --> AGE["Cosmic Age<br/>t(z) = ∫dt/a"]

    DL --> OBS["Observations<br/>Supernovae, Galaxies"]
    DA --> OBS
    AGE --> COSMO["Cosmology<br/>Ω_m, Ω_Λ, H_0"]

    style Z fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style COSMO fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

📊 Experimental Verification

1. Hubble’s Law (1929)

Observation: Galaxy spectral redshift $z\propto$ distance $d$

$$cz\approx H_{0}d$$

Hubble Constant (current): $H_0\approx 70\text{ km/s/Mpc}$

Physical Meaning: The universe is expanding.

2. Type Ia Supernovae

Standard Candle: Known absolute luminosity

Observation:

• Measure redshift $z$
• Measure apparent brightness
• Infer luminosity distance $d_L(z)$

Result (1998): Cosmic acceleration.

Nobel Prize (2011): Perlmutter, Schmidt, Riess

3. Cosmic Microwave Background (CMB)

Primordial Temperature: $T_e\approx 3000\text{ K}$ (recombination epoch)

Current Temperature: $T_0=2.725\text{ K}$

Redshift:

$$1+z=\frac{T_e}{T_0}\approx 1100$$

Agreement: $a(t_0)/a(t_e)\approx 1100$

4. Time Dilation

Supernova Light Curves:

$$\Delta t_{\text{obs}}=(1+z)\Delta t_{\text{rest}}$$

Observation (Goldhaber et al., 2001):

For supernovae at $z\sim 0.5$, light curves are indeed “stretched” by a factor of $1.5$.

Verification: Redshift is time rescaling.

graph TB
    HUBBLE["Hubble's Law<br/>z ∝ d"] --> EXP["Expanding Universe<br/>a(t)↑"]
    SN["Type Ia Supernovae<br/>Standard Candle"] --> ACC["Accelerated Expansion<br/>Λ > 0"]
    CMB["CMB Temperature<br/>T₀/T_e = 1/1100"] --> Z["Redshift<br/>z ≈ 1100"]
    TIME["Time Dilation<br/>Δt_obs = (1+z)Δt_rest"] --> VERIFY["Verification<br/>Redshift=Time Rescaling"]

    style EXP fill:#e1f5ff
    style ACC fill:#ffe1e1
    style VERIFY fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🌟 Connection to Unified Time Scale

Redshift ∈ Time Equivalence Class

Time Scale Equivalence Class:

$$[T]\sim \{\tau ,t_K,N,\lambda ,u,v,\eta ,{\omega }^{-1},z,t_{\text{mod}}\}$$

Position of Redshift:

$$1+z\sim \frac{a_0}{a_e}\sim \frac{{\kappa }_e}{{\kappa }_0}\sim \frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}$$

Affine Transformation:

$$t_{\text{cosmo}}=(1+z)t_{\text{local}}$$

Physical Meaning: Redshift is a global scaling factor of the time scale.

Connection to Phase (Chapter 1)

Recall:

$$\varphi =\frac{m{c}^2}{\hslash }\int d\tau$$

For photons ($m=0$), need to use Eikonal phase:

$$k_{\mu }={\partial }_{\mu }\varphi$$

Frequency:

$$\nu =\frac{1}{2\pi }\frac{d\varphi }{dt}$$

Redshift Formula:

$$1+z=\frac{{\nu }_e}{{\nu }_0}=\frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}$$

Theoretical Consistency: Logic of each part is mutually closed.

graph TB
    TAU["Proper Time<br/>τ"] --> PHI["Phase<br/>φ = (mc²/ℏ)∫dτ"]
    PHI --> EIKONAL["Eikonal<br/>k = ∂φ"]
    EIKONAL --> NU["Frequency<br/>ν = dφ/dt/2π"]
    NU --> Z["Redshift<br/>1+z = ν_e/ν_0"]

    Z --> KAPPA["Time Scale<br/>κ ∝ ν"]
    KAPPA --> UNIFIED["Unified Scale<br/>[T]"]

    style TAU fill:#e1f5ff
    style PHI fill:#ffe1e1
    style Z fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style UNIFIED fill:#e1ffe1,stroke:#ff6b6b,stroke-width:4px

🎓 Profound Significance

1. Relativity of Time

Special Relativity: Time dilation of moving observer $\Delta t=\gamma \Delta t_0$

General Relativity: Time dilation in gravitational field $d\tau =\sqrt{-g_{tt}}dt$

Cosmology: Time dilation from cosmic expansion $\Delta t_{\text{obs}}=(1+z)\Delta t_{\text{rest}}$

Unified View: All can be viewed as rescalings of the time scale.

2. History of the Universe

Redshift → Lookback Time:

• $z=0$: now
• $z\sim 0.1$: 1 billion years ago
• $z\sim 1$: 8 billion years ago
• $z\sim 1100$: 380,000 years after (CMB)
• $z\to \infty$: Big Bang

Redshift is the timestamp of the universe.

3. Dark Energy Mystery

Observation: At $z\sim 0.5$, universe transitions from deceleration to acceleration

Dark Energy: ${\Omega }_{\Lambda }\approx 0.7$

Cosmological Constant: $\Lambda \approx 10^{-52}{\text{ m}}^{-2}$

GLS Perspective: $\Lambda$ may be a global integration constant of the time scale (see IGVP chapter).

🤔 Exercises

1. Conceptual Understanding:

   • Why is redshift a “phase rhythm ratio”?
   • How does cosmological time dilation differ from special relativistic time dilation?
   • How does conformal time straighten null geodesics?

2. Calculation Exercises:

   • For $z=1$, calculate luminosity distance (matter-dominated universe)
   • CMB temperature $T=2.725\text{ K}$, calculate recombination temperature
   • Supernova at $z=0.5$, calculate time dilation factor

3. Observational Applications:

   • Why do different measurements of Hubble constant disagree?
   • How to infer dark energy from supernova data?
   • How does CMB power spectrum constrain cosmological parameters?

4. Advanced Thinking:

   • Redshift drift: $\dot{z}\ne 0$, can it be observed?
   • Can cosmological redshift be described using scattering theory?
   • Relationship between redshift and entropy?

Navigation:

• Previous: 06-modular-time_en.md - Modular Time
• Next: 08-time-summary_en.md - Unified Time Summary
• Overview: 00-time-overview_en.md - Unified Time Overview
• GLS Theory: unified-time-scale-geometry.md
• References:
  • Hogg, “Distance measures in cosmology” (2000)
  • Perlmutter et al., “Measurements of Ω and Λ from 42 High-Redshift Supernovae” (1999)
  • Planck Collaboration, “Planck 2018 results” (2020)