= Solution
In three dimensions an ideal Bose gas can saturate its excited-state population at <chemical potential> zero, with the remaining particles forming a <Bose-Einstein condensate>. The transition follows from the <Bose-Einstein distribution> integrated against its <density of states>; below the ideal-gas transition, $n_0/n=1-(T/T_c)^{3/2}$. Condensation alone does not supply the finite <Landau critical velocity> of an interacting superfluid: the ideal quadratic spectrum has $\inf_pE(p)/p=0$.
By contrast, a weak attractive interaction near a <Fermi surface> produces the Cooper instability and the <BCS gap equation>. Its logarithmic enhancement permits pairing at arbitrarily weak attraction in the ideal model. The low-temperature excitations have $E=\sqrt{\xi^2+\Delta^2}$, and in weak coupling $k_BT_c=(e^{\gamma_E}/\pi)\Delta(0)\simeq0.567\Delta(0)$. This transition is controlled by pair formation when phase fluctuations are small; deeply bound pairs instead motivate a Bose-condensation description with their own density and effective <mass>.
For the given ellipsoidal dispersion, set $q_x=k_x$, $q_y=k_y$ and $q_z=k_z/\sqrt\gamma$. Then $\varepsilon=\hbar^2q^2/(2m)$ while the integration measure is $d^3k=\sqrt\gamma\,d^3q$. Thus the <anisotropic ideal-gas density of states> per unit <volume> and per internal state is
$$
\boxed{N(\varepsilon,\gamma)=\sqrt\gamma\,N(\varepsilon)
=\frac{\sqrt\gamma}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt\varepsilon.}
$$
The full continuum <momentum> range is assumed; this is an anisotropic three-dimensional gas, not yet a discrete stack of two-dimensional bands.
At the Bose transition, substitute $x=\varepsilon/(k_BT)$:
$$
n_B=\frac{\sqrt\gamma}{4\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}(k_BT)^{3/2}
\int_0^\infty\frac{x^{1/2}\,dx}{e^x-1}.
$$
The integral is $\Gamma(3/2)\zeta(3/2)\simeq2.315$. Solving for the <Bose-Einstein condensation temperature for an anisotropic effective mass> gives
$$
\boxed{k_BT^{\rm BEC}=\frac{2\pi\hbar^2}{m}\left[\frac{n_B}{\sqrt\gamma\,\zeta(3/2)}\right]^{2/3}
\simeq\frac{3.31\hbar^2n_B^{2/3}}{m\gamma^{1/3}}.}
$$
The heavier $z$-direction effective <mass> increases the available excited-state count and lowers the condensation <temperature> at fixed number density.
For a balanced two-spin Fermi gas, let $n_F$ denote total <fermion> number density and take $N_s$ to be the <single-spin density of states> used in the conventional gap equation. Counting the filled ellipsoid gives
$$
\varepsilon_F(\gamma)=\frac{\hbar^2}{2m}(3\pi^2n_F)^{2/3}\gamma^{-1/3}
\equiv\varepsilon_{F0}\gamma^{-1/3},
$$
$$
N_s(\varepsilon_F,\gamma)=\frac{m(3\pi^2n_F)^{1/3}}{2\pi^2\hbar^2}\gamma^{1/3}
\equiv N_{s0}\gamma^{1/3}.
$$
If a spin-summed <density of states> is used instead, the coupling convention in the supplied integral must change accordingly; inserting a factor two into only one side would change the exponent incorrectly.
For $\varepsilon_c\ll\varepsilon_F$, replace the <density of states> across the interaction window by $N_s(\varepsilon_F,\gamma)$. Writing $\lambda_0=|g|N_{s0}$, the zero-temperature integral becomes
$$
1=\lambda_0\gamma^{1/3}\operatorname{arsinh}\left(\frac{\varepsilon_c}{\Delta}\right),
$$
and therefore
$$
\boxed{\Delta(\gamma)=\frac{\varepsilon_c}{\sinh[1/(\lambda_0\gamma^{1/3})]}
\simeq2\varepsilon_c\exp\left[-\frac1{\lambda_0\gamma^{1/3}}\right].}
$$
The second expression additionally requires $\lambda_0\gamma^{1/3}\ll1$. The explicit density dependence is $\lambda_0=|g|m(3\pi^2n_F)^{1/3}/(2\pi^2\hbar^2)$. This <mass-anisotropy enhancement of a weak-coupling BCS gap> initially raises the pairing scale, while the ideal Bose condensation scale decreases as $\gamma^{-1/3}$.
\Image[/past-exam-of-the-superconductivity-and-quantum-coherence-course-of-the-university-of-cambridge/2013/paper-1-anisotropy-scales.png]
{title=Formal anisotropy-dependent gap and Bose scales, with a schematic coherence-limited transition}
Large anisotropy weakens transport and coherence in the heavy direction. The separation of <pair formation and phase ordering> then matters, so a large mean-field gap need not imply a large actual $T_c$. A qualitative sketch has $T_c$ initially rising with the pairing scale and eventually turning down when coherence is the smaller scale. If preformed <fermion> pairs behave as <bosons>, use $n_B=n_F/2$ and pair <mass> about $2m$; in this continuum model this gives $k_BT^{\rm BEC}\simeq0.218\varepsilon_{F0}\gamma^{-1/3}$.
The turnover cannot be located reliably by taking the weak-coupling formulas to $\gamma\to\infty$: $\varepsilon_F$ falls, $\lambda_0\gamma^{1/3}$ grows, the narrow-window assumption fails and the pair <chemical potential> and <mass> require a self-consistent treatment. The smooth curve in the figure is explicitly an illustrative interpolation of two extrapolated scales, not such a calculation. If actual decoupled two-dimensional layers are approached, ordinary ideal-gas finite-temperature condensation is absent and interacting phase order requires a different analysis, including the <Berezinskii–Kosterlitz–Thouless transition> when appropriate.
The cuprates make the distinction between pairing and phase coherence relevant: they are layered and have anisotropic motion, but their strong correlations, lattice band structure and predominantly <d-wave superconductivity> are outside this attractive continuum $s$-wave model. Anisotropy by itself is therefore not an explanation or quantitative fit of cuprate transition <temperatures>.
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