The GLR-MQ equation is a nonlinear evolution equation that takes into account the shadowing effect, which tames the growth of the gluon at small-x. In this study, we analytically solve for the first time the nonlinear GLR-MQ equation using the homogeneous balance method. The definite solution of the GLR-MQ equation is obtained by fitting the MSTW2008LO gluon distribution data. We find that the geometric scaling is an inherent property of our analytical solution. Furthermore, the gluon distribution functions derived from our solution are capable of accurately reproducing the MSTW2008LO data across most regions. These results indicate that our analytical solution from the homogeneous balance method is valid to describe the gluon behavior at small-x and moderate <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup></math> regions. Moreover, the saturation scale <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>s</mi></mrow></msub></math> has been extracted from our analytical solution, we find that the energy-dependent saturation scale obeys the exponential law <math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mspace width="0.2em"></mspace><mo>∝</mo><mspace width="0.2em"></mspace><msubsup><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mi>λ</mi><mi>Y</mi></mrow></msup></math>. We also perform a numerical simulation using the analytical results at <math><msup><mrow><mi>Q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>5</mn><mspace width="0.25em"></mspace><msup><mrow><mtext>GeV</mtext></mrow><mrow><mn>2</mn></mrow></msup></math> as the initial condition. We find that our analytical results are consistent with the numerical results in small-x.

China

Yanbing Cai

Xiaopeng Wang

Xurong Chen

New analytical solution of the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu equation

We numerically study for the first time the nonlinear GLR-MQ evolution equations for nuclear parton distribution function (nPDFs) to next-to-leading order accuracy and quantify the impact of gluon recombination at small <math><mi>x</mi></math>. Using the nCTEQ15 nPDFs as input, we confirm the importance of the nonlinear corrections for small <math><mi>x</mi><mo>≲</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>, whose magnitude increases with a decrease of <math><mi>x</mi></math> and an increase of the atomic number <math><mi>A</mi></math>. We find that at <math><mi>x</mi><mo>=</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></math> and for heavy nuclei, after the upward evolution from <math><msub><mi>Q</mi><mn>0</mn></msub><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> to <math><mi>Q</mi><mo>=</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, the quark singlet <math><mi>Ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> and the gluon <math><mi>G</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> distributions become reduced by 9%–15%, respectively. The relative effect is much stronger for the downward evolution from <math><msub><mi>Q</mi><mn>0</mn></msub><mo>=</mo><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> to <math><mi>Q</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, where we find that <math><mi>Ω</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> is suppressed by 40%, while <math><mi>G</mi><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> is enhanced by 140%. These trends propagate into the <math><msubsup><mi>F</mi><mn>2</mn><mi>A</mi></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> nuclear structure function and the <math><msubsup><mi>F</mi><mi>L</mi><mi>A</mi></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> longitudinal structure function, which after the downward evolution become reduced by 45% and enhanced by 80%, respectively. Our analysis indicates that the nonlinear effects are most pronounced in <math><msubsup><mi>F</mi><mi>L</mi><mi>A</mi></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>)</mo></math> and are already quite sizable at <math><mi>x</mi><mo>∼</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> for heavy nuclei. We have checked that our conclusions very weakly depend on the choice of input nPDFs. In particular, using the EPPS21 nPDFs as input, we obtain quantitatively similar results.

Germany

J. Rausch

Finland

V. Guzey

Australia

M. Klasen

Numerical evaluation of the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu evolution equations for nuclear parton distribution functions

In this paper, we investigate the primordial perturbations of the inflation model induced from the multifield mimetic gravity, where there are two fields during inflation, and both adiabatic and isocurvature perturbation modes are generated. We show that although the original adiabatic perturbation mode indeed loses the kinetic term due to the constraint equation, by applying the curvaton mechanism where one of the fields is viewed as a curvaton field, the adiabatic perturbation can be transferred from the isocurvature one at the end of inflation. Detailed calculations are performed for both inflationary and the consequent radiation-dominant and matter-dominant epochs. Therefore, the so-called “nonpropagating problem” of the adiabatic mode will not harm the multifield mimetic inflation models.

Xin-zhe Zhang

Lei-Hua Liu

Taotao Qiu

Mimetic curvaton

Stochastic gravitational wave background (SGWB) is a promising tool to probe the very early universe where the standard model of particle physics and cosmology are connected closely. As a possible component of SGWB, gravitational waves (GW) from bubble collisions during the first order cosmological phase transitions deserve comprehensive analyses. In 2017, Ryusuke Jinno and Masahiro Takimoto proposed an elegant analysis approach to derive the analytical expressions of energy spectra of GW from bubble collisions in Minkowski spacetime avoiding large-scale numerical simulations for the first time [26]. However, they neglect the expansion of the universe and regard the duration of phase transitions as infinity in their derivation which could deviate their estimations from true values. For these two reasons, we give a new expression of GW spectra by adopting their method, switching spacetime background to FLRW spacetime, and considering a finite duration of phase transitions. By denoting σ as the fraction of the speed of phase transitions to the expansion speed of the universe, we find when σ is around   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (10), the maxima of estimated GW energy spectra drop by around 1 order of magnitude than the results given by their previous work. Even when σ = 100, the maximum of GW energy spectrum is only 65% of their previous estimation. Such a significant decrease may bring about new challenges for the detectability of GW from bubble collisions. Luckily, by comparing new spectra with PLI (power-law integrated) sensitivity curves of GW detectors, we find that the detection prospect for GW from bubble collisions is still promising for DECIGO, BBO, LISA, and TianQin in the foreseeable future.

United States

Haowen Zhong

Biping Gong

Gravitational waves from bubble collisions in FLRW spacetime

In this paper, we discuss about the possibility to enhance the tensor-to-scalar ratio r under the condition of Trans-Planckian censorship conjecture (TCC), thus <math><mi>r</mi><mo>∼</mo><mi>O</mi><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mo>)</mo></math> could be observable within the sensitivity of future experiments. We make use of the scalar-tensor theory where inflaton is nonminimally coupled to gravity. After demonstrating that the TCC condition could be modified in scalar-tensor theory, we show that due to the effects of modified gravity at the end of inflation, a large <math><mi>r</mi><mo>∼</mo><mi>O</mi><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mo>)</mo></math> could be allowed without violating the TCC. Moreover, the modification can give rise to a weak coupling of gravity to the inflation field. If such an effect has been present as early as inflation starts, it would imply that in our case, the Universe might have experienced an asymptotically safe period at its early time.

Jiaming Shi

Large against Trans-Planckian censorship in scalar-tensor theory?

The recent observations from CMB have imposed a very stringent upper-limit on the tensor/scalar ratio r of inflation models,  $$r &lt; 0.064$$  <math> <mrow> <mi>r</mi> <mo>&lt;</mo> <mn>0.064</mn> </mrow> </math>  , which indicates that the primordial gravitational waves (PGW), even though possible to be detected, should have a power spectrum of a tiny amplitude. However, current experiments on PGW is ambitious to detect such a signal by improving the accuracy to an even higher level. Whatever their results are, it will give us much information about the early Universe, not only from the astrophysical side but also from the theoretical side, such as model building for the early Universe. In this paper, we are interested in analyzing what kind of inflation models can be favored by future observations, starting with a kind of general action offered by the effective field theory (EFT) approach. We show a general form of r that can be reduced to various models, and more importantly, we show how the accuracy of future observations can put constraints on model parameters by plotting the contours in their parameter spaces.

Taishi Katsuragawa

Shulei Ni

Confronting inflation models with the coming observations on primordial gravitational waves

Motivated by the problem of expanding the single-trace tree-level amplitude of Einstein-Yang-Mills theory to the BCJ basis of Yang-Mills amplitudes, we present an alternative expansion formula in gauge invariant vector space. Starting from a generic vector space consisting of polynomials of momenta and polarization vectors, we define a new sub-space as a gauge invariant vector space by imposing constraints on the gauge invariant conditions. To characterize this sub-space, we compute its dimension and construct an explicit gauge invariant basis from it. We propose an expansion formula in this gauge invariant basis with expansion coefficients being linear combinations of the Yang-Mills amplitude, manifesting the gauge invariance of both the expansion basis and coefficients. With the help of quivers, we compute the expansion coefficients via differential operators and demonstrate the general expansion algorithm using several examples.

Bo Feng

Xiao-Di Li

Rijun Huang

Expansion of EYM amplitudes in gauge invariant vector space B.F. is supported by Qiu-Shi Funding and the National Natural Science Foundation of China (NSFC) (11935013, 11575156); R.H. is supported by the National Natural Science Foundation of China (NSFC) (11805102), Natural Science Foundation of Jiangsu Province (BK20180724) and ShuangChuang Talent Program of Jiangsu Province

In this paper, we investigate the potential-driven inflation models with a disformal coupling to Einstein gravity, to find out the effects of such a coupling on these models. We consider a simple coupling form which introduces only one parameter, and three inflation models, namely the chaotic inflation, the Higgs inflation, and the monodromy inflation. We find that the disformal coupling can have some modifications to the observational variables of these models such as the power spectrum, the spectral index as well as the tensor/scalar ratio, although not too large due to the constraints on the disformal coupling parameter. With these modifications, one has the opportunity of improving models that lie on the edge of the favorable regions of Planck observational data, such as monodromy inflation. Moreover, the nontrivial sound speed of tensor perturbations (gravitational waves) may come out, due to the coupling of gravity and kinetic terms of the field.

Zehua Xiao

Muhsin Aljaf

Potential-driven inflation with disformal coupling to gravity

We study the chameleon field dark matter, dubbed scalaron, in F (R) gravity in the Big Bang Nucleosynthesis (BBN) epoch. With an R 2-correction term required to solve the singularity problem for F (R) gravity, we first find that the scalaron dynamics is governed by the R 2 term and the chameleon mechanism in the early universe, which makes the scalaron physics model-independent regarding the low-energy scale modification. In viable F (R) dark energy models including the R 2 correction, our analysis suggests the scalaron universally evolves in a way with a bouncing oscillation irrespective of the low-energy modification for the late-time cosmic acceleration. Consequently, we find a universal bound on the scalaron mass in the BBN epoch, to be reflected on the constraint for the coupling strength of the R 2 term, which turns out to be more stringent than the one coming from the fifth force experiments. It is then shown that the scalaron naturally develops a small enough fluctuation in the BBN epoch, hence can avoid the current BBN constraint placed by the latest Planck 2018 data, and can also have a large enough sensitivity to be hunted by the BBN, with more accurate measurements for light element abundances as well as the baryon number density fraction.

Hua Chen

Shinya Matsuzaki

Big Bang Nucleosynthesis hunts chameleon dark matter

In this work, we revisit the perturbations that are generated in the bounce inflation scenario constructed within the framework of f(T) theory. It has been well known that pure f(T) theory cannot give rise to bounce inflation behavior, so aside from the gravity part, we also employ a canonical scalar field for minimal extension. We calculate the perturbations in f(T) theory using the well-established ADM formalism, and find various conditions to avoid their pathologies. We find that it is indeed very difficult to obtain a healthy model without those pathologies, however, one may find a way out if a potential requirement, say, to keep every function continuous, is abandoned.