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• W3215733235 abstract "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 $$beta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>β</mml:mi> </mml:math> -times $$t_beta :=int ^t a^{-2beta }$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>β</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mo>∫</mml:mo> <mml:mi>t</mml:mi> </mml:msup> <mml:msup> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>β</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems $$begin{aligned} O_{1/2} Psi =frac{Lambda }{12}Psi , quad O_1 a =-frac{Lambda }{3} a , end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mi>Ψ</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>Λ</mml:mi> <mml:mn>12</mml:mn> </mml:mfrac> <mml:mi>Ψ</mml:mi> <mml:mo>,</mml:mo> <mml:mspace /> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mi>Λ</mml:mi> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> which is suggestive of a measurement problem. $$O_{beta }(rho ,p)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>β</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The $$O_beta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> ’s are also independent of the spatial curvature, labeled by k , and absorbed in $$begin{aligned} Psi =sqrt{a} e^{frac{i}{2}sqrt{k}eta } . end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mi>a</mml:mi> </mml:msqrt> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mfrac> <mml:mi>i</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msqrt> <mml:mi>k</mml:mi> </mml:msqrt> <mml:mi>η</mml:mi> </mml:mrow> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$eta equiv t_{1/2}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>≡</mml:mo> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> among the $$t_beta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>β</mml:mi> </mml:msub> </mml:math> ’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." @default.
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• W3215733235 date "2021-12-01" @default.
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• W3215733235 title "Universe as Klein–Gordon eigenstates" @default.
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