Universe as Klein–Gordon eigenstates
Abstract We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the $$\beta $$ β -times $$t_\beta :=\int ^t a^{-2\beta }$$ t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it t...
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
SpringerOpen
2021
|
| Materias: | |
| Acceso en línea: | https://doaj.org/article/c445670e1eda45c2b0767ed5c2ddd36c |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| id |
oai:doaj.org-article:c445670e1eda45c2b0767ed5c2ddd36c |
|---|---|
| record_format |
dspace |
| spelling |
oai:doaj.org-article:c445670e1eda45c2b0767ed5c2ddd36c2021-12-05T12:09:17ZUniverse as Klein–Gordon eigenstates10.1140/epjc/s10052-021-09865-41434-60441434-6052https://doaj.org/article/c445670e1eda45c2b0767ed5c2ddd36c2021-12-01T00:00:00Zhttps://doi.org/10.1140/epjc/s10052-021-09865-4https://doaj.org/toc/1434-6044https://doaj.org/toc/1434-6052Abstract We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the $$\beta $$ β -times $$t_\beta :=\int ^t a^{-2\beta }$$ t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems $$\begin{aligned} O_{1/2} \Psi =\frac{\Lambda }{12}\Psi , \quad O_1 a =-\frac{\Lambda }{3} a , \end{aligned}$$ O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. $$O_{\beta }(\rho ,p)$$ O β ( ρ , p ) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The $$O_\beta $$ O β ’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} \Psi =\sqrt{a} e^{\frac{i}{2}\sqrt{k}\eta } . \end{aligned}$$ Ψ = a e i 2 k η . The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ k = 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$\eta \equiv t_{1/2}$$ η ≡ t 1 / 2 among the $$t_\beta $$ t β ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space.Marco MatoneSpringerOpenarticleAstrophysicsQB460-466Nuclear and particle physics. Atomic energy. RadioactivityQC770-798ENEuropean Physical Journal C: Particles and Fields, Vol 81, Iss 12, Pp 1-6 (2021) |
| institution |
DOAJ |
| collection |
DOAJ |
| language |
EN |
| topic |
Astrophysics QB460-466 Nuclear and particle physics. Atomic energy. Radioactivity QC770-798 |
| spellingShingle |
Astrophysics QB460-466 Nuclear and particle physics. Atomic energy. Radioactivity QC770-798 Marco Matone Universe as Klein–Gordon eigenstates |
| description |
Abstract We formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the $$\beta $$ β -times $$t_\beta :=\int ^t a^{-2\beta }$$ t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems $$\begin{aligned} O_{1/2} \Psi =\frac{\Lambda }{12}\Psi , \quad O_1 a =-\frac{\Lambda }{3} a , \end{aligned}$$ O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. $$O_{\beta }(\rho ,p)$$ O β ( ρ , p ) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The $$O_\beta $$ O β ’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} \Psi =\sqrt{a} e^{\frac{i}{2}\sqrt{k}\eta } . \end{aligned}$$ Ψ = a e i 2 k η . The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ k = 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$\eta \equiv t_{1/2}$$ η ≡ t 1 / 2 among the $$t_\beta $$ t β ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space. |
| format |
article |
| author |
Marco Matone |
| author_facet |
Marco Matone |
| author_sort |
Marco Matone |
| title |
Universe as Klein–Gordon eigenstates |
| title_short |
Universe as Klein–Gordon eigenstates |
| title_full |
Universe as Klein–Gordon eigenstates |
| title_fullStr |
Universe as Klein–Gordon eigenstates |
| title_full_unstemmed |
Universe as Klein–Gordon eigenstates |
| title_sort |
universe as klein–gordon eigenstates |
| publisher |
SpringerOpen |
| publishDate |
2021 |
| url |
https://doaj.org/article/c445670e1eda45c2b0767ed5c2ddd36c |
| work_keys_str_mv |
AT marcomatone universeaskleingordoneigenstates |
| _version_ |
1718372217743474688 |