On the Hubble tension

 This problem is another instance of us observing history. We have a problem whose solution might be very close, or very far. It is like those textbook stories in history of physics that a simple function needs to be found; in this case \(E(z)\) defined here.

There are many factors at work in determining \(E(z)\) and some necessary information may not yet be available. The wisest approach seems to be trying to constrain the form of \(E(z)\) based on sound theoretical considerations. 

The simplest interpretation of the Hubble tension is to say that we must somehow re-define Hubble parameter such that $$H(z)=E(z)\bar{H}(z),$$ where \(\bar{H}(z)\) is the `current' function. Thinking about the definition of Hubble parameter there are three possibilities that can account for the Hubble tension:

1. There is yet to be found a differential equation governing \(H\) that makes this modification possible;
2. We must change the scale factor;
3. We must change the derivative in the definition

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de Sitter universe

$$z+1=e^{-Ht}$$ $$t=-\frac{\ln(z+1)}{H}$$

$$\tilde{H}=H\big(1+(1+q)\ln(z+1)\big)$$

$$\boxed{E(z)=1+(1+q)\ln(z+1)}$$