Understanding the Age of the Universe: Five Minutes with High-School Physics

Understanding the Age of the Universe: Five Minutes with High-School Physics Alexandre Amalric Columbia University ORCID: 0009-0000-0685-5868 December 16, 2025 Abstract Where do we come from? This short technical note derives a closed-form estimate of the actual consensus age of the universe, which is about 13.8billion years. I only use high-school physics : Newtonian mechanics and energy conservation. My goal is to show how a seemingly complex cosmological question can be made transparent through simple reasoning and clear exposition. Contents 1 How Astronomers Measure Cosmic Expansion 1 2 High-School Physics Energy Formula 2 3 Simple Uniform Sphere Universe Model 3 4 Rewinding Cosmic Time to Big Bang 3 5 Conclusion: Numerical Estimate 3 1 How Astronomers Measure Cosmic Expansion Thanks to the Hubble Space Telescope, astronomers can measure how fast the universe expands and thus quantify the Hubble constant H0from which we can determine the Hubble time tH≡1 H0 ≈14 billion years,(1) a first “ballpark” estimate of the universe’s age. A cosmic model would give us the theoretical time of the universe t0=C×1 H0 ,(2) using Cis a dimensionless number that depends on cosmic ingredients. My goal is to derive Cusing only high-school physics. 1 2 High-School Physics Energy Formula (Annotated for first-time readers) The high-school mechanical constant energy under gravity is E=1 2m v2− G |{z} Newton’s constant m M |{z} mass of the universe R |{z} size of the universe ,(3) with v≡dR/dt the expansion speed of the universe. Reading the two right-hand-side terms: The first term is the kinetic energy needed to drive expansion whereas the second is the gravity that resists it. E=1 2m v2−G m M R.(4) No external “engine” in our universe choice for the energy constant. To model an expansion explained only by the matter in our universe (self-gravitating balance corresponds to the Einstein–de Sitter universe (1932)) we take the minimal-energy case with E= 0: 0 = 1 2m v2−G m M R.(5) The reasoning above is deliberately written in a very familiar, high-school style, using the mechanical energy of a test mass mso that standard textbook formulas can be applied directly. A more rigorous formulation would instead work with the specific energy (energy per unit mass) of the universe, which is also conserved. Since the test mass mplays no physical role in the cosmic dynamics, we therefore divide by m immediately: 1 2v2−GM R= 0.(6) Then 1 2v2=GM R=⇒v R2 =2GM R3.(7) Define H(t)the Hubble instantaneous expansion rate of the universe H(t)≡v R,(8) to obtain the Newtonian (high-school version of) Friedmann form H2=2GM R3.(9) (If you want to compare with the actual complex relativistic Friedmann equations...) 2 3 Simple Uniform Sphere Universe Model We will model the entire universe as one big, uniform sphere of radius R. The volume of this sphere (measured in cubic meters (m3)) is given by V=4 3πR3.(10) To obtain the total mass Mof the sphere in kilograms (kg), we multiply by the density ρin kilograms per cubic meter (kg m−3), so that the units of volume cancel (kg m−3m3= kg) and this gives the total mass: M=4 3πR3ρ. (11) In this “cosmic sphere”, the enclosed mass remains constant over time t: M(t) = 4 3π R3(t)ρ(t) = M0=constant.(12) As R(t)increases (expansion), the density must fall so that M(t)still equals M0: 4 3π R3(t)ρ(t) = 4 3π R3 0ρ0⇒ρ(t)=ρ0R0 R(t)3 .(13) Substitute M0=4 3πρ0R3 0into (9): H2=2G R34 3πρ0R3 0=8πG 3ρ0R0 R3 .(14) 4 Rewinding Cosmic Time to Big Bang Using H=v/R = (1/R)dR/dt, (14) becomes after taking the square root (power one half) 1 R dR dt =r8πG 3ρ0R0 R3/2 ⇐⇒ R1/2dR =r8πG 3ρ0R3/2 0dt. (15) Integrate explicitly from the bang (R=0 at t=0) to today (R0at t0): ZR0 0 R1/2dR =r8πG 3ρ0R3/2 0Zt0 0 dt ⇒2 3R3/2 0=r8πG 3ρ0R3/2 0t0.(16) Cancel R3/2 0and use H2 0=8πG 3ρ0: t0=2 3H0 .(17) We have just derived the well-known "C= 2/3law" of the Einstein–de Sitter benchmark. 5 Conclusion: Numerical Estimate Let’s use the Hubble constant measured with the Hubble Space Telescope, H0= 70 km s−1Mpc−1= 2.27 ×10−18 s−1, [this is an astronomer unit... 1 Mpc=3.0857 ×1022 m.] t0=2 3H0 ≈9.3billion years.(18) 3 Estimated age: t0≈10 billion years Final remark. For a first-principles, high-school-physics derivation, this result is fair enough. For your information, the actual consensus value of this 1 H0Cconstant comes from the full relativistic Lambda Cold Dark Matter (ΛCDM) model, which includes matter and dark energy: t0=1 H0Z∞ 0 dz (1 + z |{z} redshift )sΩm |{z} matter (1+z)3+ ΩΛ |{z} dark energy drives cosmic acceleration , These additional parameters make the “cosmic clock” of this advanced model tick faster, hence an older consensus age of the universe of 13.8 billion years compared to our 10 billion year estimate. Appendix Modern cosmological age measurement. •Planck Collaboration (2018), “Planck 2018 results. I. Overview and the cosmological legacy of Planck,” Astronomy & Astrophysics 641, A1. See Table of cosmological parameters, which reports t0= 13.797 ±0.023 Gyr.PDF / DOI link Einstein–de Sitter & the 2/3factor. •A. Einstein & W. de Sitter (1932), On the Relation between the Expansion and the Mean Density of the Universe, Proc. Natl. Acad. Sci. 18, 213. DOI:10.1073/pnas.18.3.213 •University lecture notes deriving a(t)∝t2/3⇒t0= 2/(3H0)(see e.g. Wikipedia summary). Friedmann–Lemaître background. •A. Friedmann (1922), Über die Krümmung des Raumes, Z. Phys. 10, 377. DOI:10.1007/BF01332580 •G. Lemaître (1931 English), A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extragalactic nebulae, MNRAS 91, 483. DOI:10.1093/mnras/91.5.483 •Relativistic context: Friedmann equations (overview). 4