Universe as Klein-Gordon Eigenstates
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866917834009870336 |
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| author | Matone, Marco |
| author_facet | Matone, Marco |
| contents | We formulate Friedmann's equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the $β$-times $t_β:=\int^t a^{-2β}$, where $a$ is the scale factor. In particular, it turns out that Friedmann's equations are equivalent to the eigenvalue problems $$ O_{1/2} Ψ=\fracΛ{12}Ψ\ , \qquad O_1 a =\fracΛ{3} a \ , $$ which is suggestive of a measurement problem. $O_β(ρ,p)$ are space-independent Klein-Gordon operators, depending only on energy density and pressure, and related to the Klein-Gordon Hamilton-Jacobi equations. The $O_β$'s are also independent of the spatial curvature, labeled by $k$, and absorbed in $$ Ψ=\sqrt a e^{\frac{i}{2}\sqrt{k}η} \ . $$ The above pair of equations is the unique possible linear form of Friedmann's equations unless $k=0$, in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $η\equiv t_{1/2}$ among the $t_β$'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 | Preprint |
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arxiv_https___arxiv_org_abs_2110_01557 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Universe as Klein-Gordon Eigenstates Matone, Marco High Energy Physics - Theory Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Phenomenology Quantum Physics We formulate Friedmann's equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the $β$-times $t_β:=\int^t a^{-2β}$, where $a$ is the scale factor. In particular, it turns out that Friedmann's equations are equivalent to the eigenvalue problems $$ O_{1/2} Ψ=\fracΛ{12}Ψ\ , \qquad O_1 a =\fracΛ{3} a \ , $$ which is suggestive of a measurement problem. $O_β(ρ,p)$ are space-independent Klein-Gordon operators, depending only on energy density and pressure, and related to the Klein-Gordon Hamilton-Jacobi equations. The $O_β$'s are also independent of the spatial curvature, labeled by $k$, and absorbed in $$ Ψ=\sqrt a e^{\frac{i}{2}\sqrt{k}η} \ . $$ The above pair of equations is the unique possible linear form of Friedmann's equations unless $k=0$, in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $η\equiv t_{1/2}$ among the $t_β$'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. |
| title | Universe as Klein-Gordon Eigenstates |
| topic | High Energy Physics - Theory Cosmology and Nongalactic Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Phenomenology Quantum Physics |
| url | https://arxiv.org/abs/2110.01557 |