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In an attractor with one adiabatic clock, the leading squeezed primordial bispectrum is the response to a coordinate dilation. Holding a halo's physical mass and smoothing scale fixed cancels this dilation in its abundance response. Thus the consistency-relation term does not generate the physical $k^{-2}$ scale-dependent halo bias from local non-Gaussianity. For example, a top-hat filter of fixed physical radius has coordinate radius $R_c=e^{-{\zeta }_L}{R}_0$, while the short density contrast is ${\delta }_s(\mathbf{x})={\delta }_s^{(0)}(e^{{\zeta }_L}\mathbf{x})$. The substitution $\mathbf{y}=e^{{\zeta }_L}\mathbf{x}$ shows that its normalized average over $|\mathbf{x}|<R_c$ equals the original average over $|\mathbf{y}|<R_0$. Genuine local effects start at two spatial derivatives of the long mode. Non-attractor backgrounds or nonstandard initial states need not satisfy this argument.

Normalize the linear growth factor to $D(0)=1$. The collapse overdensity at the collapse epoch is ${\delta }_{\mathrm{sc}}\simeq 1.686$ for the matter-dominated spherical-collapse model. The present-extrapolated spherical-collapse barrier isIt is the initial linear overdensity, extrapolated to today, required to collapse by $t$; it is not the nonlinear density contrast of a virialized halo. The smoothed matter density variance ${\sigma }^2(M,z)$ is the variance of the linear density contrast smoothed on a Lagrangian comoving scale containing mass $M$. For a spherical top-hat filter,Consequently $\sigma (M,z)=D(z)\sigma (M,0)$ for scale-independent linear growth. The equivalent halo peak height conventions are $\nu ={\delta }_c(t)/\sigma (M,0)={\delta }_{\mathrm{sc}}/\sigma (M,z)$; using both an evolved barrier and an evolved variance would count growth twice.

To calculate a number-density growth time, use a narrow fixed-mass bin, not the collapsed mass fraction itself. Differentiate the Press-Schechter formalism mass fraction and divide the mass density in the resulting interval by $M$. This gives the Press-Schechter halo mass functionAt fixed $M$, the mass factor and logarithmic slope do not depend on time. Since $\dot{\nu }/\nu =-\dot{D}/D$, the Press-Schechter abundance growth at fixed mass isThe prefactor $\nu$ matters here: the exponential-only rare-peak approximation drops the minus one and is accurate only for $\nu \gg 1$.

Read the supplied variance relation as a mass-scale calibration using the numerical velocity label given for the selected population. It gives $\sigma (M,3)=0.85\sqrt{250/10}=4.25$. Matter domination between the two high-redshift epochs gives $D(19)/D(3)\simeq 4/20$, so $\sigma (M,19)\simeq 0.85$ and $\nu \simeq 1.984$. At the epoch in question,Thus the requested fixed-mass-bin growth estimate with that calibration isAn exponential-only approximation gives about $8.1\times 10^7\mathrm{yr}$, which is somewhat shorter because this is only a roughly two-sigma population.

The wording leaves two sample conventions worth distinguishing. First, an actual virial velocity of a spherical-overdensity halo is epoch-dependent at fixed mass. If the variance fit is instead calibrated using physical virial velocities at redshift three, the same mass whose velocity is $10\mathrm{km}{\mathrm{s}}^{-1}$ at redshift nineteen has $v(3)\simeq 10\sqrt{4/20}=4.47\mathrm{km}{\mathrm{s}}^{-1}$. The halo virial-velocity conversion between epochs then gives $\sigma (M,19)\simeq 1.271$, $\nu \simeq 1.327$ and $t_d\simeq 4.2\times 10^8\mathrm{yr}$ for a fixed-mass bin. The two numerical answers reflect the velocity-label convention in the supplied fit, not two ways of differentiating one fixed fit.

Second, a cumulative number density is $n(>M)={\int }_M^{\infty }(dn/d{M}^{\prime })d{M}^{\prime }$, not simply ${\bar{\rho }}_{m,0}f(>M)/M$: the latter is a mass fraction divided by a threshold mass, not the number of objects. The cumulative derivative is an abundance-weighted average of $(\nu (M^{\prime }{)}^2-1)\dot{D}/D$ over that integral. If the supplied power-law variance fit is extended over all larger masses, with the same velocity-label convention, direct integration gives a cumulative-number growth time of about $8.4\times 10^7\mathrm{yr}$ instead. A sample maintained at fixed physical velocity at successive epochs additionally moves its mass boundary and needs a selection convention. The numerical estimate above explicitly uses the ordinary fixed-mass differential interpretation. These distinctions are important when an exact growth time rather than a rare-tail estimate is intended.

For an attractor with one adiabatic clock and a regular Bunch-Davies vacuum, a long comoving curvature perturbation ${\zeta }_L$ acts at leading order as a spatial dilation. The single-field inflation squeezed-limit consistency relation therefore fixesFor the local convention $\zeta ={\zeta }_G+(3/5)f_{\mathrm{NL}}({\zeta }_G^2-⟨{\zeta }_G^2⟩)$, comparison with its squeezed primordial bispectrum givesThe matter-era negative gravitational potential is $\phi =3\zeta /5$, so this is also the conventional $f_{\mathrm{NL}}$ of part (i). In single-field slow-roll inflation, the spectral tilt is small and the effective local coupling is of slow-roll size, typically of order $10^{-2}$ for a tilt of a few percent. This is a statement about the normalized coupling: the squeezed primordial bispectrum still contains the large long-mode power spectrum.

Formally putting this small coupling into the local template of part (ii) predicts only a tiny correction on accessible scales, rather than an order-unity local signal. For a physical halo abundance there is a stronger qualification. The consistency-relation modulation at fixed coordinate wavelength is a dilation; the coordinate size of a fixed physical halo must be dilated too. Those responses cancel. This is the absence of primordial scale-dependent halo bias in single-clock inflation: the leading consistency-relation term produces no physical $k^{-2}$ halo bias. The first physical long-mode response contains two spatial derivatives, so dividing it by ${\delta }_L\propto k_L^2{\phi }_L$ does not produce the local-template enhancement. Thus the leading primordial large-scale galaxy power spectrum remains the Gaussian-bias result, apart from ordinary transfer, bias, and projection effects.

The cancellation can be seen with a top-hat filter. At fixed local time, a constant long curvature rescales physical lengths by $e^{{\zeta }_L}$, so the coordinate radius of a fixed physical halo is $R_c=e^{-{\zeta }_L}{R}_0$, and the coordinate short density contrast is ${\delta }_s(\mathbf{x})={\delta }_s^{(0)}(e^{{\zeta }_L}\mathbf{x})$. Changing variables $\mathbf{y}=e^{{\zeta }_L}\mathbf{x}$ givesThus the smoothed field and its variance at fixed physical mass are unchanged by the constant long mode.

The assumptions matter: a non-attractor such as ultra-slow-roll inflation, additional fluctuating fields, or an excited initial state need not obey this squeezed-limit conclusion. A sizable physical local modulation would test the stated single-clock attractor assumptions, rather than every possible one-field model.

The spherical-collapse model applies a spherical top-hat filter to an overdense region and predicts collapse when its linearly extrapolated smoothed overdensity exceeds the threshold ${\delta }_c\simeq 1.686$ in an Einstein-de Sitter universe.