Graduate Hubble scale and energy density during inflation

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The discussion focuses on the relationship between energy density and the Hubble scale during inflation, specifically questioning the formula H^4 = (9/64π^2)(ρ^2/M_P^4). Participants explore how the energy density, represented as ρ = 3M_Pl^2H^2 for a flat universe, is modified during the inflationary period. Clarifications are sought on the implications of this relationship and how it affects cosmological models. Resources are requested for a deeper understanding of these concepts in the context of inflationary theory. The conversation emphasizes the need for precise formulations to accurately describe the early universe dynamics.
spaghetti3451
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Is the energy density ##\rho## in the early universe and the Hubble scale ##H## during inflation related by the formula

$$H^{4} = \frac{9}{64\pi^{2}} \frac{\rho^{2}}{M_{P}^{4}}?$$

Here, ##M_{P}## is the Planck mass.

Can you suggest any resources where this is explained?
 
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Let me rephrase my question.

The energy density for a spatially flat universe (without a cosmological constant) is ##\rho = 3 M_{\text{Pl}}^{2}H^{2}.##

How is this formula modified during inflation?
 

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