I have 2 time series, X_t and Y_t, which are on different scales.

Y_t can be 0 to infinite, while X_t is limited to 0 to 100.

How can I determine if the trend of X_t forecast the trend of Y_t? In other words if there is a peak in Xt, then the peak of Yt will follow after some lag. If this is indeed the case, what is the lag? I am not interested in forecasting the actual value of Yt.

Using the following chart as an illustration, the red line is Xt (which in my data the values are between 27 to 34), and the black line is Yt (which is about 40000).

I tried to use Time Lagged Pearson Correlation, but I am aware the pearson correlation (of the 2 time series) does not have the concept of time. Pearson correlation simply treats the time series as lists of data.

I have read some guides on Granger causality, but it seems this checks if (the value of) Xt is useful in forecasting the value of Yt, which is similar to a regression framework. (which I am mostly interested in forecasting the trend of Yt)

I am a newbie in time series analysis, Thanks for your time!