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We know the size of observable universe, that it is expanding, that expansion is accelerating. Does sufficiently precise data exist to determine at least the SIGN of the next (the 3rd) derivative? If Dark Energy is causing acceleration, shouldn't expansion dilute the repulsive force of dark energy? While continuing to accelerate, shouldn't the rate of acceleration diminish over time? Shouldn't $\frac{{d^3}s}{dt^3}$ be negative? If were again surprised and determine that the acceleration itself is increasing, then maybe we're not exploding, maybe we're being pulled apart.

I'll leave theory to others to ponder. I just want to know when we can do some more High School math to determine the next, the 3rd, derivative of the size of the universe? Does the data exist?

Wouldn't it be nice to know?

Kyle Kanos
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William Schwartz
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1 Answers1

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If we take the simple approach of determining the state of the "jerk" today by assuming an exponential expansion (e.g., $a(t)\sim\exp(H_0 t)$), then $$ \dot a=H_0a\tag{1} $$ The derivative of this is then, $$ \frac{d^2a}{dt^2}=H_0\dot{a}=H^2_0a $$ And now for the "jerk," $$ \frac{d^3a}{dt^3}=H^2_0\dot{a}=H^3_0a\tag{2} $$ The Hubble constant is already pretty small at about 70 km/s/Mpc (2.26$\cdot$10$^{-18}$ 1/s), so taking the cube of this makes for a very small number and is probably a difficult value to tack down observationally. Since the scale factor is non-negative, then we can conclude that (2) is also non-negative, at least for now

Adding the cosmological constants (and neglecting the expected 0 values) such that (1) becomes $$ \dot{a}=H_0a\sqrt{\Omega_{M,0}a^{-3} + \Omega_{\Lambda,0}}\tag{3} $$ probably won't change the sign of the result, given that both $\Omega_{M,0}$ and $\Omega_{\Lambda,0}$ are positive constants.

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Kyle Kanos
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