Is the raised Levi-Civita symbol a tensor density of weight 1?

Question

In Sean Carroll's GR book, pg 83, between eqs. (2.69-70), the Levi-Civita symbol with raised indices is defined as
$$tilde{epsilon}^{mu_1mu_2...mu_n}=text{sgn}(g)tilde{epsilon}_{mu_1 mu_2...mu_n}$$
where $text{sgn}(g) = frac{g}{|g|}$ is the sign function of the metric determinant $g$ and $tilde{epsilon}_{mu_1 mu_2...mu_n}$ is the Levi-Civita symbol we usually use (which is a tensor density of weight $1$).
He then said that the Levi-Civita symbol $tilde{epsilon}^{mu_1mu_2...mu_n}$ is a tensor density of weight $-1$.
I tried to prove his statement:
$$begin{equation}
tilde{epsilon}^{mu_1'mu_2'...mu_n'}=frac{g'}{|g'|}tilde{epsilon}_{mu_1' mu_2'...mu_n'} 
= frac{|frac{partial x^{mu'}}{partial x^mu}| ^{-2} g} {|frac{partial x^{mu'}}{partial x^mu}| ^{-2} |g| } | frac{partial x^{mu'}}{partial x^mu}|  tilde{epsilon}_{mu_1 mu_2...mu_n} frac{partial x^{mu_1}}{partial x^{{mu}'_1}}frac{partial x^{mu_2}}{partial x^{mu_2'}}...frac{partial x^{mu_n}}{partial x^{mu_n'}}
=  |frac{partial x^{mu'}}{partial x^mu}|frac{g}{|g|}tilde{epsilon}_{mu_1 mu_2...mu_n} frac{partial x^{mu_1}}{partial x^{{mu}'_1}}frac{partial x^{mu_2}}{partial x^{mu_2'}}...frac{partial x^{mu_n}}{partial x^{mu_n'}}
= |frac{partial x^{mu'}}{partial x^mu}| tilde{epsilon}^{mu_1mu_2...mu_n}frac{partial x^{mu_1}}{partial x^{{mu}'_1}}frac{partial x^{mu_2}}{partial x^{mu_2'}}...frac{partial x^{mu_n}}{partial x^{mu_n'}}
end{equation}$$
This shows that $tilde{epsilon}^{mu_1mu_2...mu_n}$ is a tensory density of weight $1$. What am I doing wrong?

J. Murray · Accepted Answer

To me, a more conceptually direct derivation goes as follows. The Levi-Civita symbol is a symbol, not a geometrical object.  We can denote its value in a coordinate system $x$ as $tilde epsilon_{(x)munurhosigma}$.  On its face, this is silly - the value of $tilde epsilon$ does not depend on the coordinate system. However, the point here is that we can compare the (trivial) transformation behavior of $tildeepsilon$ with the transformation behavior we would expect from a $(0,4)$-tensor.
As stated, the actual transformation behavior of these components is trivial, i.e.
$$tilde epsilon_{(x)munurhosigma}mapsto tilde epsilon_{(y)munurhosigma} = tilde epsilon_{(x)munurhosigma}$$
If it were a tensor, then it would transform as
$$tildeepsilon_{(x)munurhosigma}mapsto tilde epsilon_{(y)munurhosigma} = tilde epsilon_{(x)alphabetagammadelta}J^alpha_{  mu}J^beta_{  nu}J^gamma_{  rho}J^delta_{  sigma}= J tildeepsilon_{(x)munurhosigma}$$
where $J^alpha_{  beta} equiv frac{partial x^alpha}{partial y^beta}$ is the Jacobian matrix and $J$ is the Jacobian determinant.  Therefore, we say that the true transformation behavior is given by
$$tildeepsilon_{(x)munurhosigma}mapsto tilde epsilon_{(y)munurhosigma} = J^{-1}tilde epsilon_{(x)alphabetagammadelta}J^alpha_{  mu}J^beta_{  nu}J^gamma_{  rho}J^delta_{  sigma}$$
The fact that the power of $J$ out front is $-1$ means that $tilde epsilon$ is a tensor density of weight $-1$.

We now turn our attention to the symbol with upstairs indices.  In this case, we note that the actual transformation behavior is now
$$tilde epsilon_{(x)}^{munusigmarho}mapsto tilde epsilon_{(y)}^{munusigmarho} = mathrm{sgn}(J)epsilon_{(x)}^{munusigmarho}$$
whereas its expected transformation behavior (if it were a tensor) would be
$$tilde epsilon_{(x)}^{munusigmarho}mapsto tilde epsilon_{(y)}^{munusigmarho} =epsilon_{(x)}^{alphabetagammadelta}(J^{-1})^mu_{  alpha}(J^{-1})^nu_{  beta}(J^{-1})^rho_{  gamma}(J^{-1})^sigma_{  delta}=J^{-1} tildeepsilon_{(x)}^{munurhosigma}$$
Therefore, we can write that the true transformation behavior is
$$tilde epsilon_{(x)}^{munurhosigma} mapsto tilde epsilon_{(y)}^{munurhosigma} = mathrm{sgn}(J) Jepsilon_{(x)}^{alphabetagammadelta}(J^{-1})^mu_{  alpha}(J^{-1})^nu_{  beta}(J^{-1})^rho_{  gamma}(J^{-1})^sigma_{  delta}$$
The fact that the power of $J$ is now $+1$ means that the weight is now $+1$; the fact that we also have a $mathrm{sgn}(J)$ means that $tilde epsilon^{munurhosigma}$ is in fact a pseudotensor density of weight $+1$.  Carroll does not discuss this because he assumes that the coordinate transformation in question is orientation-preserving, which implies that $mathrm{sgn}(J)=+1$ and the distinction between tensors and pseudotensors (and their respective densities) can be ignored.

In summary, we contrast the actual transformation behavior of the $tilde epsilon$ symbols with their expected transformation behavior if they were genuine tensors to determine in what way the actual transformation behavior differs.  If it transforms like a tensor except for an additional factor of $J^w$, we call it a tensor density of weight $w$.  If it additional picks up a factor of $mathrm{sgn}(J)$, we call it a pseudotensor density.

Is the raised Levi-Civita symbol a tensor density of weight 1?

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