A structure theorem for 2-stretched Gorenstein algebras

In this paper we study the isomorphism classes of local, Artinian, Gorenstein k-algebras A whose maximal ideal M satisfies dim_k(M^3/M^4)=1 by means of Macaulay's inverse system generalizing a recent result by J. Elias and M.E. Rossi. Then we use such results in order to complete the description of the singular locus of the Gorenstein locus of the punctual Hilbert scheme of degree 11.Comment: 24 pages. We removed lemma 2.1 because it was false and we modified the proof of proposition 3.2 accordingly inserting some new due reference

Dissipative Axial Inflation

We analyze in detail the background cosmological evolution of a scalar field coupled to a massless abelian gauge field through an axial term $\frac{\phi}{f_\gamma} F \tilde{F}$, such as in the case of an axion. Gauge fields in this case are known to experience tachyonic growth and therefore can backreact on the background as an effective dissipation into radiation energy density $\rho_R$, which which can lead to inflation without the need of a flat potential. We analyze the system, for momenta $k$ smaller than the cutoff $f_\gamma$, including numerically the backreaction. We consider the evolution from a given static initial condition and explicitly show that, if $f_\gamma$ is smaller than the field excursion $\phi_0$ by about a factor of at least ${\cal O} (20)$, there is a friction effect which turns on before that the field can fall down and which can then lead to a very long stage of inflation with a generic potential. In addition we find superimposed oscillations, which would get imprinted on any kind of perturbations, scalars and tensors. Such oscillations have a period of 4-5 efolds and an amplitude which is typically less than a few percent and decreases linearly with $f_\gamma$. We also stress that the comoving curvature perturbation on uniform density should be sensitive to slow-roll parameters related to $\rho_R$ rather than $\dot{\phi}^2/2$, although we postpone a calculation of the power spectrum and of non-gaussianity to future work and we simply define and compute suitable slow roll parameters. Finally we stress that this scenario may be realized in the axion case, if the coupling $1/f_\gamma$ to U(1) (photons) is much larger than the coupling $1/f_G$ to non-abelian gauge fields (gluons), since the latter sets the range of the potential and therefore the maximal allowed $\phi_0\sim f_G$.Comment: 22 pages, 27 figure

Examples of rank two aCM bundles on smooth quartic surfaces in P3\mathbb{P}^3

Let $F\subseteq\mathbb{P}^3$ be a smooth quartic surface and let $\mathcal{O}_F(h):=\mathcal{O}_{\mathbb{P}^3}(1)\otimes\mathcal{O}_F$. In the present paper we classify locally free sheaves $\mathcal{E}$ of rank $2$ on $F$ such that $c_1(\mathcal{E})=\mathcal{O}_F(2h)$, $c_2(\mathcal{E})=8$ and $h^1\big(F,\mathcal{E}(th)\big)=0$ for $t\in\mathbb{Z}$. We also deal with their stability.Comment: 22 pages. Exposition improve

Thermalized Axion Inflation

We analyze the dynamics of inflationary models with a coupling of the inflaton $\phi$ to gauge fields of the form $\phi F \tilde{F}/f$, as in the case of axions. It is known that this leads to an instability, with exponential amplification of gauge fields, controlled by the parameter $\xi= \dot{\phi}/(2fH)$, which can strongly affect the generation of cosmological perturbations and even the background. We show that scattering rates involving gauge fields can become larger than the expansion rate $H$, due to the very large occupation numbers, and create a thermal bath of particles of temperature $T$ during inflation. In the thermal regime, energy is transferred to smaller scales, radically modifying the predictions of this scenario. We thus argue that previous constraints on $\xi$ are alleviated. If the gauge fields have Standard Model interactions, which naturally provides reheating, they thermalize already at $\xi\gtrsim2.9$, before perturbativity constraints and also before backreaction takes place. In absence of SM interactions (i.e. for a dark photon), we find that gauge fields and inflaton perturbations thermalize if $\xi\gtrsim3.4$; however, observations require $\xi\gtrsim6$, which is above the perturbativity and backreaction bounds and so a dedicated study is required. After thermalization, though, the system should evolve non-trivially due to the competition between the instability and the gauge field thermal mass. If the thermal mass and the instabilities equilibrate, we expect an equilibrium temperature of $T_{eq} \simeq \xi H/\bar{g}$ where $\bar{g}$ is the effective gauge coupling. Finally, we estimate the spectrum of perturbations if $\phi$ is thermal and find that the tensor to scalar ratio is suppressed by $H/(2T)$, if tensors do not thermalize.Comment: 36 pages, 6 figures, Published versio

Higgs Mass and Gravity Waves in Standard Model False Vacuum Inflation

In previous publications we have proposed that Inflation can be realized in a second minimum of the Standard Model Higgs potential at energy scales of about $10^{16}$ GeV, if the minimum is not too deep and if a mechanism which allows a transition to the radiation dominated era can be found. This is provided, {\it e.g.}, by scalar-tensor gravity models or hybrid models. Using such ideas we had predicted the Higgs boson mass to be of about $126\pm 3$ GeV, which has been confirmed by the LHC, and that a possibly measurable amount of gravity waves should be produced. Using more refined recent theoretical calculations of the RGE we show that such scenario has the right scale of Inflation only for small Higgs mass, lower than about 124 GeV, otherwise gravity waves are overproduced. The precise value is subject to some theoretical error and to experimental errors on the determination of the strong coupling constant. Such an upper bound corresponds also to the recent claimed measurement by BICEP2 of the scale of inflation through primordial tensor modes. Finally we show that introducing a moderately large non-minimal coupling for the Higgs field the bound can shift to larger values and be reconciled with the LHC measurements of the Higgs mass.Comment: 6 pages, 4 figure

Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\leq 13$ and find its components when $d = 14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of $\mathbb{P}^n$ for $d\geq 4$.Comment: v4: final. v2: expanded proof of Theorems A and B. 33 pages, comments welcome

CMB Aberration and Doppler Effects as a Source of Hemispherical Asymmetries

Our peculiar motion with respect to the CMB rest frame represents a preferred direction in the observed CMB sky since it induces an apparent deflection of the observed CMB photons (aberration) and a shift in their frequency (Doppler). Both effects distort the multipoles $a_{\ell m}$'s at all $\ell$'s. Such effects are real as it has been recently measured for the first time by Planck according to what was forecast in some recent papers. However, the common lore when estimating a power spectrum from CMB is to consider that Doppler affects only the $\ell=1$ multipole, neglecting any other corrections. In this work we use simulations of the CMB sky in a boosted frame with a peculiar velocity $\beta = v/c = 1.23 \times 10^{-3}$ in order to assess the impact of such effect on power spectrum estimations in different regions of the sky. We show that the boost induces a north-south asymmetry in the power spectrum which is highly significant and non-negligible, of about (0.58 $\pm$ 0.10)% for half-sky cuts when going up to $\ell$ = 2500. We suggest that these effects are relevant and may account for some of the north-south asymmetries seen in the Planck data, being especially important at small scales. Finally we analyze the particular case of the ACT experiment, which observed only a small fraction of the sky and show that it suffers a bias of about 1% on the power spectrum and of similar size on some cosmological parameters: for example the position of the peaks shifts by 0.5% and the overall amplitude of the spectrum is about 0.4% lower than a full-sky case.Comment: 13 pages, 5 figure

Standard Model False Vacuum Inflation: Correlating the Tensor-to-Scalar Ratio to the Top Quark and Higgs Boson masses

For a narrow band of values of the top quark and Higgs boson masses, the Standard Model Higgs potential develops a false minimum at energies of about $10^{16}$ GeV, where primordial Inflation could have started in a cold metastable state. A graceful exit to a radiation-dominated era is provided, e.g., by scalar-tensor gravity models. We pointed out that if Inflation happened in this false minimum, the Higgs boson mass has to be in the range $126.0 \pm 3.5$ GeV, where ATLAS and CMS subsequently reported excesses of events. Here we show that for these values of the Higgs boson mass, the inflationary gravitational wave background has be discovered with a tensor-to-scalar ratio at hand of future experiments. We suggest that combining cosmological observations with measurements of the top quark and Higgs boson masses represents a further test of the hypothesis that the Standard Model false minimum was the source of Inflation in the Universe.Comment: v1: 4 pages, 2 figures; v2: 5 pages, 2 figures, improvements in the text; v3: 5 pages, 2 figures, minor improvements in the text, matches PRL versio